Permanent tags

• 0000 Chapter Błocki, Zbigniew; Kołodziej, Sławomir. On regularization of plurisubharmonic functions on manifolds. Proc. Amer. Math. Soc. 135 (2007), no. 7, 2089–2093.
• 0001 Chapter Błocki, Zbigniew. The Calabi-Yau theorem. Complex Monge-Ampère equations and geodesics in the space of Kähler metrics, 201–227, Lecture Notes in Math., 2038, Springer, Heidelberg, 2012.
• 0002 Chapter Boucksom, Sébastien; Jonsson, Mattias. Tropical and non-Archimedean limits of degenerating families of volume forms. J. Éc. polytech. Math. 4 (2017), 87–139.
• 0003 Chapter Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias. Solution to a non-Archimedean Monge-Ampère equation. J. Amer. Math. Soc. 28 (2015), no. 3, 617–667.
• 0004 Chapter Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias. The non-Archimedean Monge-Ampère equation. Nonarchimedean and tropical geometry, 31–49, Simons Symp., Springer, [Cham], 2016.
• 0005 Chapter Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias. Singular semipositive metrics in non-Archimedean geometry. J. Algebraic Geom. 25 (2016), no. 1, 77–139.
• 0006 Chapter Caffarelli, L. A. A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. of Math. (2) 131 (1990), no. 1, 129–134.
• 0007 Chapter Caffarelli, Luis A. Interior W2,pW^{2,p} estimates for solutions of the Monge-Ampère equation. Ann. of Math. (2) 131 (1990), no. 1, 135–150.
• 0008 Chapter Caffarelli, Luis A. A note on the degeneracy of convex solutions to Monge Ampère equation. Comm. Partial Differential Equations 18 (1993), no. 7-8, 1213–1217.
• 0009 Chapter Caffarelli, Luis A.; Viaclovsky, Jeff A. On the regularity of solutions to Monge-Ampère equations on Hessian manifolds. Comm. Partial Differential Equations 26 (2001), no. 11-12, 2339–2351.
• 000A Chapter Chambert-Loir, Antoine. Heights and measures on analytic spaces. A survey of recent results, and some remarks. Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume II, 1–50, London Math. Soc. Lecture Note Ser., 384, Cambridge Univ. Press, Cambridge, 2011.
• 000B Chapter Chambert-Loir, Antoine; Ducros, Antoine. Formes différentielles réelles et courants sur les espaces de Berkovich. arXiv:1204.6277.
• 000C Chapter Cheeger, Jeff; Naber, Aaron. Regularity of Einstein manifolds and the codimension 4 conjecture. Ann. of Math. (2) 182 (2015), no. 3, 1093–1165.
• 000D Chapter Chan, Kwokwai. The Strominger-Yau-Zaslow conjecture and its impact. Selected Expository Works of Shing-Tung Yau with Commentary. Vol. II, 1183-1208, Adv. Lect. Math. (ALM) 29, Int. Press, Somerville, MA, 2014
• 000E Chapter Chen, Xiuxiong; Cheng, Jingrui. On the constant scalar curvature Kähler metrics (I)—A priori estimates. J. Amer. Math. Soc. 34 (2021), no. 4, 909–936.
• 000F Chapter Tristan C. Collins, Yang Li. Complete Calabi-Yau metrics in the complement of two divisors. arXiv:2203.10656.
• 000G Chapter Collins, Tristan C.; Tosatti, Valentino. An extension theorem for Kähler currents with analytic singularities. Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 4, 893–905.
• 000H Chapter Chen, Huayi; Moriwaki, Atsushi. Extension property of semipositive invertible sheaves over a non-archimedean field. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 1, 241–282.
• 000I Chapter Coman, Dan; Guedj, Vincent; Zeriahi, Ahmed. Extension of plurisubharmonic functions with growth control. J. Reine Angew. Math. 676 (2013), 33–49.
• 000J Chapter Demailly, Jean-Pierre; Pali, Nefton. Degenerate complex Monge-Ampère equations over compact Kähler manifolds. Internat. J. Math. 21 (2010), no. 3, 357–405.
• 000K Chapter De Giorgi, Ennio. Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961.
• 000L Chapter Donaldson, Simon K. Kähler geometry on toric manifolds, and some other manifolds with large symmetry. Handbook of geometric analysis. No. 1, 29–75, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, 2008.
• 000M Chapter Donaldson, Simon; Sun, Song. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. Acta Math. 213 (2014), no. 1, 63–106.
• 000N Chapter Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed. Singular Kähler-Einstein metrics. J. Amer. Math. Soc. 22 (2009), no. 3, 607–639.
• 000P Chapter Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed. A priori L∞L^{\infty}-estimates for degenerate complex Monge-Ampère equations. Int. Math. Res. Not. IMRN 2008, Art. ID rnn 070, 8 pp.
• 000Q Chapter Fang, Yanbo. Non-Archimedean metric extension for semipositive line bundles. arXiv:1904.03696.
• 000R Chapter Foscolo, Lorenzo. ALF gravitational instantons and collapsing Ricci-flat metrics on the K​3K3 surface. J. Differential Geom. 112 (2019), no. 1, 79–120.
• 000S Chapter Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín. Arithmetic geometry of toric varieties. Metrics, measures and heights. Astérisque No. 360, (2014), vi+222 pp.
• 000T Chapter Gross, Mark. Mirror symmetry and the Strominger-Yau-Zaslow conjecture. Current developments in mathematics 2012, 133–191, Int. Press, Somerville, MA, 2013.
• 000U Chapter Gross, Mark. Topological mirror symmetry. Invent. Math. 144 (2001), no. 1, 75–137.
• 000V Chapter Gross, Mark; Tosatti, Valentino; Zhang, Yuguang. Collapsing of abelian fibered Calabi-Yau manifolds. Duke Math. J. 162 (2013), no. 3, 517–551.
• 000W Chapter Gross, Mark; Wilson, P. M. H. Large complex structure limits of K​3K3 surfaces. J. Differential Geom. 55 (2000), no. 3, 475–546.
• 000X Chapter Guedj, Vincent; Zeriahi, Ahmed. Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15 (2005), no. 4, 607–639.
• 000Y Chapter Gubler, Walter. Forms and current on the analytification of an algebraic variety (after Chambert-Loir and Ducros). Nonarchimedean and tropical geometry, 1–30, Simons Symp., Springer, [Cham], 2016.
• 000Z Chapter Gubler, Walter; Jell, Philipp; Künnemann, Klaus; Martin, Florent. Continuity of plurisubharmonic envelopes in non-archimedean geometry and test ideals. With an appendix by José Ignacio Burgos Gil and Martín Sombra. Ann. Inst. Fourier (Grenoble) 69 (2019), no. 5, 2331–2376.
• 0010 Chapter Gubler, Martin Gubler, Walter; Martin, Florent. On Zhang’s semipositive metrics. Doc. Math. 24 (2019), 331–372.
• 0011 Chapter Haase, Christian; Zharkov, Ilia. Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I. arXiv:math/0205321.
• 0012 Chapter Haase, Christian; Zharkov, Ilia. Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces II. arXiv:math/0301222.
• 0013 Chapter Harvey, Reese; Lawson, H. Blaine, Jr. Calibrated geometries. Acta Math. 148 (1982), 47–157.
• 0014 Chapter Hein, Hans-Joachim; Sun, Song; Viaclovsky, Jeff; Zhang, Ruobing. Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces. arXiv:1807.09367.
• 0015 Chapter Joyce, Dominic. Singularities of special Lagrangian fibrations and the SYZ conjecture. Comm. Anal. Geom. 11 (2003), no. 5, 859–907.
• 0016 Chapter Joyce, Dominic D. Riemannian holonomy groups and calibrated geometry. Oxford Graduate Texts in Mathematics, 12. Oxford University Press, Oxford, 2007. x+303 pp. ISBN: 978-0-19-921559-1
• 0017 Chapter Joyce, Dominic. Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. EMS Surv. Math. Sci. 2 (2015), no. 1, 1–62.
• 0018 Chapter Kempf, G.; Knudsen, Finn Faye; Mumford, D.; Saint-Donat, B. Toroidal embeddings. I. Lecture Notes in Mathematics, Vol. 339. Springer-Verlag, Berlin-New York, 1973. viii+209 pp.
• 0019 Chapter Kołodziej, Sławomir. The complex Monge-Ampère equation. Acta Math. 180 (1998), no. 1, 69–117
• 001A Chapter Kołodziej, Sławomir. The Monge-Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52 (2003), no. 3, 667–686.
• 001B Chapter Kołodziej, Sławomir. Hölder continuity of solutions to the complex Monge-Ampère equation with the right-hand side in LpL^{p}: the case of compact Kähler manifolds. Math. Ann. 342 (2008), no. 2, 379–386.
• 001C Chapter Kontsevich, Maxim; Soibelman, Yan. Homological mirror symmetry and torus fibrations. Symplectic geometry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publ., River Edge, NJ, 2001.
• 001D Chapter Kontsevich, Maxim; Soibelman, Yan. Affine structures and non-Archimedean analytic spaces. The unity of mathematics, 321–385, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006.
• 001E Chapter Kontsevich, Maxim; Tschinkel, Yuri. Non-archimedean Kähler geometry. Unpublished note, 2002.
• 001F Chapter Li, Y. Uniform Skoda integrability and Calabi-Yau degeneration. arXiv:2006.16961.
• 001G Chapter Li, Y. SYZ conjecture for Calabi-Yau hypersurfaces in the Fermat family. arXiv:1912.02360. accepted by Acta. Math.
• 001H Chapter Li, Y. Metric SYZ conjecture and non-archimedean geometry. arXiv:2007.01384.
• 001I Chapter Li, Yang. SYZ geometry for Calabi-Yau 3-folds: Taub-NUT and Ooguri-Vafa type metrics. arXiv:1902.08770. accepted by AMS Memoir.
• 001J Chapter Li, Yang. PhD thesis, Imperial College London (2019).
• 001K Chapter Li, Y; Tosatti, Valentino. Diameter bounds for degenerating Calabi-Yau metrics. accepted by JDG.
• 001L Chapter Li, Y. Thomas-Yau conjecture and holomorphic curves. arXiv:2203.01467.
• 001M Chapter Matessi, Diego; Castaño Bernard, Ricardo. Lagrangian 3-torus fibrations. J. Differential Geom. 81 (2009), no. 3, 483–573.
• 001N Chapter Enrica Mazzon, Léonard Pille-Schneider. Toric geometry and integral affine structures in non-archimedean mirror symmetry. arXiv:2110.04223.
• 001P Chapter Mikhalkin, Grigory. Decomposition into pairs-of-pants for complex algebraic hypersurfaces. Topology 43 (2004), no. 5, 1035–1065.
• 001Q Chapter Mooney, Connor. Partial regularity for singular solutions to the Monge-Ampère equation. Comm. Pure Appl. Math. 68 (2015), no. 6, 1066–1084.
• 001R Chapter Nicaise, Johannes; Xu, Chenyang. The essential skeleton of a degeneration of algebraic varieties. Amer. J. Math. 138 (2016), no. 6, 1645–1667.
• 001S Chapter Nicaise, Johannes; Xu, Chenyang; Yu, Tony Yue. The non-archimedean SYZ fibration. Compos. Math. 155 (2019), no. 5, 953–972.
• 001T Chapter Odaka, Yuji; Oshima, Yoshiki. Collapsing K3 surfaces and Moduli compactification. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 8, 81–86.
• 001U Chapter Savin, Ovidiu. Small perturbation solutions for elliptic equations. Comm. Partial Differential Equations 32 (2007), no. 4-6, 557–578.
• 001V Chapter Siu, Yum Tong. Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics. DMV Seminar, 8. Birkhäuser Verlag, Basel, 1987. 171 pp. ISBN: 3-7643-1931-3
• 001W Chapter Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric. Mirror symmetry is TT-duality. Nucl.Phys.B479:243-259,1996.
• 001X Chapter Sun, Song; Zhang, Ruobing. Complex structure degenerations and collapsing of Calabi-Yau metrics. arXiv:1906.03368.
• 001Y Chapter Tong, Freid; Guo, Bin; Phong, D.H. Stability estimates for the complex Monge-Ampère and Hessian equations. arXiv:2106.03913.
• 001Z Chapter Tong, Freid; Guo, Bin; Phong, D.H. On L∞L^{\infty} estimates for complex Monge-Ampère equations. arXiv:2106.02224.
• 0020 Chapter Tosatti, Valentino. Limits of Calabi-Yau metrics when the Kähler class degenerates. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 4, 755–776.
• 0021 Chapter Tosatti, Valentino. Adiabatic limits of Ricci-flat Kähler metrics. J. Differential Geom. 84 (2010), no. 2, 427–453.
• 0022 Chapter Tian, Gang. On Kähler-Einstein metrics on certain Kähler manifolds with C1​(M)>0C_{1}(M)>0. Invent. Math. 89 (1987), no. 2, 225–246.
• 0023 Chapter Thomas, R. P. Moment maps, monodromy and mirror manifolds. Symplectic geometry and mirror symmetry (Seoul, 2000), 467–498, World Sci. Publ., River Edge, NJ, 2001.
• 0024 Chapter Thomas, R. P.; Yau, S.-T. Special Lagrangians, stable bundles and mean curvature flow. Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113.
• 0025 Chapter Vilsmeier, Christian. A comparison of the real and non-archimedean Monge-Ampère operator. Math. Z. 297 (2021), no. 1-2, 633–668.
• 0026 Chapter Yau, Shing Tung. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.
• 0027 Chapter Zharkov, Ilia. Limiting behavior of local Calabi-Yau metrics. Adv. Theor. Math. Phys. 8 (2004), no. 3, 395–420.
• 0028 Chapter Zeriahi, Ahmed. Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions. Indiana Univ. Math. J. 50 (2001), no. 1, 671–703.
• 0029 Chapter Zhang, Yuguang. Collapsing of Calabi-Yau manifolds and special Lagrangian submanifolds. Univ. Iagel. Acta Math. No. 54 (2017), 53–78.
• 002A Chapter Survey on the metric SYZ conjecture and non-archimedean geometry
• 002B Acknowledgement Acknowledgement .
• 002C Conjecture Conjecture 2.1 .
• 002D Notation Notation .
• 002E Question Question 1 .
• 002F Question Question 2 .
• 002G Question Question 3 .
• 002H Remark Remark 1 .
• 002I Question Question 4 .
• 002J Remark Remark 2 .
• 002K Example Example 3.1 .
• 002L Example Example 3.2 .
• 002M Question Question 5 .
• 002N Remark Remark 3 .
• 002P Theorem Theorem 4.1 .
• 002Q Remark Remark 4 .
• 002R Theorem Theorem 4.2 .
• 002S Remark Remark 5 .
• 002T Theorem Theorem 4.3 .
• 002U Theorem Theorem 4.4 .
• 002V Remark Remark 6 .
• 002W Remark Remark 7 .
• 002X Theorem Theorem 4.5 .
• 002Y Theorem Theorem 4.6 .
• 002Z Theorem Theorem 4.7 .
• 0030 Theorem Theorem 4.8 .
• 0031 Remark Remark 8 .
• 0032 Remark Remark 9 .
• 0033 Definition Definition 5.1 .
• 0034 Remark Remark 10 .
• 0035 Proposition Proposition 5.2 .
• 0036 Proposition Proposition 5.3 .
• 0037 Remark Remark 11 .
• 0038 Proposition Proposition 5.4 .
• 0039 Remark Remark 12 .
• 003A Theorem Theorem 5.5 .
• 003B Definition Definition 5.6 .
• 003C Remark Remark 13 .
• 003D Question Question 6 .
• 003E Theorem Theorem 6.1 .
• 003F Theorem Theorem 6.2 .
• 003G Proposition Proposition 6.3 .
• 003H Proof Proof.
• 003I Remark Remark 14 .
• 003J Remark Remark 15 .
• 003K Proposition Proposition 6.4 .
• 003L Proof Proof.
• 003M Proposition Proposition 6.5 .
• 003N Remark Remark 16 .
• 003P Question Question 7 .
• 003Q Lemma Lemma 6.6 .
• 003R Proof Proof.
• 003S Proposition Proposition 6.7 .
• 003T Proof Proof.
• 003U Remark Remark 17 .
• 003V Question Question 8 .
• 003W Remark Remark 18 .
• 003X Proposition Proposition 6.8 .
• 003Y Lemma Lemma 6.9 .
• 003Z Section Overview
• 0040 Section Metric SYZ conjecture
• 0041 Subsection The genesis of the SYZ conjecture
• 0042 Subsection Further motivations
• 0043 Subsection Collapsing K3 surfaces with elliptic surfaces
• 0044 Subsection Best hope on Calabi-Yau 3-folds
• 0045 Subsection Strong vs. weak SYZ conjecture
• 0046 Section Large complex structure limit
• 0047 Subsection Volume asymptote and essential skeleton
• 0048 Subsection The effect of blow up
• 0049 Subsection Kontsevich-Soibelman conjecture
• 004A Section Analytical foundations
• 004B Subsection Yau’s solution to the Calabi conjecture
• 004C Subsection Complex pluripotential theory
• 004D Subsection Skoda inequality
• 004E Subsection Estimate on pluripotentials
• 004F Subsection Savin’s small perturbation theorem
• 004G Subsection Regularity theory for real Monge-Ampère
• 004H Subsection Special Lagrangian fibration
• 004I Section Nonarchimedean geometry
• 004J Subsection Berkovich space, hybrid topology
• 004K Subsection Model functions, metrics, positivity
• 004L Subsection Approximation by Fubini-Study metrics
• 004M Subsection NA Monge-Ampère measure
• 004N Subsection NA Calabi conjecture
• 004P Subsection Comparison property
• 004Q Section Glimpse of proof strategy
• 004R Subsection Reduction to potential estimates
• 004S Subsection Strategy I: non-archimedean geometry
• 004T Subsection Motivation for NA geometry
• 004U Subsection Grafting the real MA solution
• 004V Subsection C 0 -convergence of the potential
• 004W Subsection Strategy II: a priori limit
• 004X Subsection Producing convex functions
• 004Y Subsection C 0 -convergence of the potential and extension problem
• 004Z Subsection Real MA metric
• 0050 Subsection Relation to NA geometry
• 0051 Equation CalabiYaumeasureeqn
• 0052 Equation measureconvergence
• 0053 Equation Lpdensity
• 0054 Equation Skodaassumption
• 0055 Equation SlagZhang
• 0056 Equation Calabiconjecture1
• 0057 Equation hybridconvergence
• 0058 Equation FubiniStudyNA
• 0059 Equation NAMACY
• 005A Equation realMACY
• 005B Bibliography Blocki
• 005C Bibliography Blockilecture
• 005D Bibliography Boucksom1
• 005E Bibliography Boucksom
• 005F Bibliography Boucksomsurvey
• 005G Bibliography Boucksomsemipositive
• 005H Bibliography Caff1
• 005I Bibliography Caff2
• 005J Bibliography Caff4
• 005K Bibliography CaffarelliViaclovsky
• 005L Bibliography ChambertLoir
• 005M Bibliography CLDucros
• 005N Bibliography CheegerNaber
• 005P Bibliography Chan
• 005Q Bibliography ChenCheng
• 005R Bibliography CollinsLi
• 005S Bibliography CollinsTosatti
• 005T Bibliography ChenMoriwaki
• 005U Bibliography Coman
• 005V Bibliography DemaillyPali
• 005W Bibliography DeGiorgi
• 005X Bibliography Donaldsontoric
• 005Y Bibliography DonaldsonSun
• 005Z Bibliography EGZ
• 0060 Bibliography EGZ2
• 0061 Bibliography Fang
• 0062 Bibliography Lorenzo
• 0063 Bibliography Gil
• 0064 Bibliography Gross
• 0065 Bibliography Gross2
• 0066 Bibliography GrossTosattiZhang
• 0067 Bibliography GrossWilson
• 0068 Bibliography GZ
• 0069 Bibliography Gubler
• 006A Bibliography GublerJill
• 006B Bibliography GublerMartin
• 006C Bibliography HaaseZharkov
• 006D Bibliography HaaseZharkov2
• 006E Bibliography HarveyLawson
• 006F Bibliography HSVZ
• 006G Bibliography Joyce
• 006H Bibliography Joycebook
• 006I Bibliography Joyceconj
• 006J Bibliography ToroidalembeddingsI
• 006K Bibliography Kolodziej
• 006L Bibliography KolodziejL1
• 006M Bibliography KolodziejHolder
• 006N Bibliography KS2
• 006P Bibliography KS
• 006Q Bibliography KontsevichTschinkel
• 006R Bibliography LiSkoda
• 006S Bibliography LiFermat
• 006T Bibliography LiNA
• 006U Bibliography LiTaub
• 006V Bibliography Lithesis
• 006W Bibliography LiTosatti
• 006X Bibliography LiThomasYau
• 006Y Bibliography MatessiLag
• 006Z Bibliography Leonard
• 0070 Bibliography Mikhalkin
• 0071 Bibliography Mooney
• 0072 Bibliography NicaiseXu
• 0073 Bibliography NicaiseXuYu
• 0074 Bibliography Odaka
• 0075 Bibliography Savin
• 0076 Bibliography Siu
• 0077 Bibliography SYZ
• 0078 Bibliography HSVZ2
• 0079 Bibliography Tong1
• 007A Bibliography Tong2
• 007B Bibliography Tosattidiameter
• 007C Bibliography Tosatti
• 007D Bibliography Tian
• 007E Bibliography Thomas
• 007F Bibliography ThomasYau
• 007G Bibliography Vilsmeier
• 007H Bibliography Yau
• 007I Bibliography Zharkov
• 007J Bibliography Zeriahi
• 007K Bibliography Zhang
• 007L Theorem Theorem 1.1 .
• 007M Remark Remark 1.2 .
• 007N Theorem Theorem 1.3 .
• 007P Theorem Theorem 1.4 .
• 007Q Remark Remark 1.5 .
• 007R Acknowledgement Acknowledgement .
• 007S Lemma Lemma 2.1 .
• 007T Proof Proof.
• 007U Lemma Lemma 2.2 .
• 007V Proof Proof.
• 007W Lemma Lemma 2.3 .
• 007X Proof Proof.
• 007Y Proposition Proposition 2.4 .
• 007Z Proof Proof.
• 0080 Remark Remark 2.5 .
• 0081 Lemma Lemma 2.6 .
• 0082 Proof Proof.
• 0083 Proposition Proposition 2.7 .
• 0084 Corollary Corollary 2.8 .
• 0085 Theorem Theorem 2.9 .
• 0086 Proof Proof.
• 0087 Proof Proof.
• 0088 Theorem Theorem 3.1 .
• 0089 Section Introduction
• 008A Section Uniform Skoda inequality
• 008B Subsection Quantitative stratification and good test functions
• 008C Subsection Convexity
• 008D Subsection Harnack type inequality
• 008E Subsection Local L 1 estimate
• 008F Subsection Local Skoda estimate
• 008G Subsection Uniform global Skoda estimate
• 008H Section Application to Calabi-Yau degeneration
• 008I Subsection Calabi-Yau measure
• 008J Subsection Uniform Skoda estimate
• 008K Subsection Uniform L ∞ -estimate
• 008L Equation Skodaassumption
• 008M Equation CalabiYaumeasureeqn
• 008N Equation measureupperbound
• 008P Bibliography Blockilecture
• 008Q Bibliography Boucksomsemipositive
• 008R Bibliography Boucksom1
• 008S Bibliography Boucksom
• 008T Bibliography Boucksomsurvey
• 008U Bibliography DemaillyPali
• 008V Bibliography Eleonora
• 008W Bibliography EGZ
• 008X Bibliography EGZ2
• 008Y Bibliography GrossTosattiZhang
• 008Z Bibliography GZ
• 0090 Bibliography ToroidalembeddingsI
• 0091 Bibliography KS2
• 0092 Bibliography KS
• 0093 Bibliography Li
• 0094 Bibliography LiSYZ
• 0095 Bibliography LiTosatti
• 0096 Bibliography SYZ
• 0097 Bibliography Tian
• 0098 Bibliography Tosatti
• 0099 Bibliography Yau
• 009A Bibliography Zeriahi
• 009B Conjecture Conjecture 1.1 .
• 009C Remark Remark 1.2 .
• 009D Theorem Theorem 1.3 .
• 009E Notation Notation .
• 009F Acknowledgement Acknowledgement .
• 009G Theorem Theorem 2.1 .
• 009H Theorem Theorem 2.2 .
• 009I Corollary Corollary 2.3 .
• 009J Lemma Lemma 2.4 .
• 009K Lemma Lemma 2.5 .
• 009L Proof Proof.
• 009M Theorem Theorem 2.6 .
• 009N Proof Proof.
• 009P Remark Remark 2.7 .
• 009Q Theorem Theorem 2.8 .
• 009R Remark Remark 2.9 .
• 009S Remark Remark 3.1 .
• 009T Definition Definition 3.2 .
• 009U Remark Remark 3.3 .
• 009V Remark Remark 3.4 .
• 009W Proposition Proposition 3.5 .
• 009X Remark Remark 3.6 .
• 009Y Proposition Proposition 3.7 .
• 009Z Remark Remark 3.8 .
• 00A0 Remark Remark 3.9 .
• 00A1 Theorem Theorem 3.10 .
• 00A2 Definition Definition 3.11 .
• 00A3 Proposition Proposition 3.12 .
• 00A4 Remark Remark 3.13 .
• 00A5 Lemma Lemma 4.1 .
• 00A6 Proof Proof.
• 00A7 Lemma Lemma 4.2 .
• 00A8 Proof Proof.
• 00A9 Proposition Proposition 4.3 .
• 00AA Proof Proof.
• 00AB Proposition Proposition 4.4 .
• 00AC Proof Proof.
• 00AD Remark Remark 4.5 .
• 00AE Corollary Corollary 4.6 .
• 00AF Proof Proof.
• 00AG Theorem Theorem 4.7 .
• 00AH Proof Proof.
• 00AI Remark Remark 4.8 .
• 00AJ Lemma Lemma 4.9 .
• 00AK Proof Proof.
• 00AL Theorem Theorem 4.10 .
• 00AM Proof Proof.
• 00AN Theorem Theorem 4.11 .
• 00AP Remark Remark 4.12 .
• 00AQ Remark Remark 5.1 .
• 00AR Example Example 5.2 .
• 00AS Example Example 5.3 .
• 00AT Example Example 5.4 .
• 00AU Remark Remark 5.5 .
• 00AV Remark Remark 5.6 .
• 00AW Remark Remark 5.7 .
• 00AX Section Introduction
• 00AY Section Analytic backgrounds
• 00AZ Subsection Uniform Skoda inequality
• 00B0 Subsection Kolodziej’s estimate on pluripotentials
• 00B1 Subsection L 1 -stability estimate
• 00B2 Subsection Savin’s small perturbation theorem
• 00B3 Subsection Regularity theory for real Monge-Ampère
• 00B4 Section Nonarchimedean geometry
• 00B5 Subsection Volume asymptote and essential skeleton
• 00B6 Subsection Berkovich space, hybrid topology
• 00B7 Subsection Model functions, metrics, positivity
• 00B8 Subsection NA Monge-Ampère measure
• 00B9 Subsection NA Calabi conjecture
• 00BA Subsection Approximation by Fubini-Study metrics
• 00BB Section Potential estimates and SYZ fibration
• 00BC Subsection Comparison Kähler metric I
• 00BD Subsection Comparison Kähler metric II: Regularisation
• 00BE Subsection Potential estimate I
• 00BF Subsection Potential estimate II
• 00BG Subsection Metric convergence and SYZ fibration
• 00BH Section Further directions
• 00BI Subsection Transcendental case
• 00BJ Subsection Conjectural meaning of the NA MA measure I
• 00BK Subsection Conjectural meaning of NA MA II
• 00BL Subsection Non-maximal degenerations, generalised Calabi ansatz
• 00BM Equation CalabiYaumeasureeqn
• 00BN Equation Skodaassumption
• 00BP Equation FubiniStudyXt
• 00BQ Equation measureconvergence
• 00BR Equation hybridconvergence
• 00BS Equation NAMACY
• 00BT Equation realMACY
• 00BU Equation FubiniStudyNA
• 00BV Equation CYmetricasymptote
• 00BW Equation H11class
• 00BX Equation curvatureformasymptote
• 00BY Equation measureasymptotegeneralisedCY
• 00BZ Equation NAmeasureformula
• 00C0 Equation NAMAPDE
• 00C1 Equation generalisedCalabifibre
• 00C2 Equation generalizedCalabimetric
• 00C3 Bibliography Blockilecture
• 00C4 Bibliography Boucksomsemipositive
• 00C5 Bibliography Boucksom1
• 00C6 Bibliography Boucksom
• 00C7 Bibliography Boucksomsurvey
• 00C8 Bibliography Boucksomnew1
• 00C9 Bibliography Boucksomnew2
• 00CA Bibliography Caff1
• 00CB Bibliography Caff2
• 00CC Bibliography Caff4
• 00CD Bibliography ChambertLoir
• 00CE Bibliography CLDucros
• 00CF Bibliography ChenMoriwaki
• 00CG Bibliography DemaillyPali
• 00CH Bibliography EGZ
• 00CI Bibliography EGZ2
• 00CJ Bibliography Fang
• 00CK Bibliography GrossTosattiZhang
• 00CL Bibliography GrossWilson
• 00CM Bibliography Gubler
• 00CN Bibliography GublerJill
• 00CP Bibliography GublerMartin
• 00CQ Bibliography GZ
• 00CR Bibliography HarveyLawson
• 00CS Bibliography HSVZ
• 00CT Bibliography Joyce
• 00CU Bibliography ToroidalembeddingsI
• 00CV Bibliography KS2
• 00CW Bibliography KS
• 00CX Bibliography KolodziejL1
• 00CY Bibliography Li
• 00CZ Bibliography LiFermat
• 00D0 Bibliography LiuniformSkoda
• 00D1 Bibliography LiTosatti
• 00D2 Bibliography Mooney
• 00D3 Bibliography NicaiseXu
• 00D4 Bibliography NicaiseXuYu
• 00D5 Bibliography Odaka
• 00D6 Bibliography RongZhang
• 00D7 Bibliography Savin
• 00D8 Bibliography SYZ
• 00D9 Bibliography HSVZ2
• 00DA Bibliography Tosattidiameter
• 00DB Bibliography Tosatti
• 00DC Bibliography Vilsmeier
• 00DD Bibliography Yau
• 00DE Bibliography Zharkov
• 00DF Bibliography Zeriahi
• 00DG Bibliography Zhang
• 00DH Bibliography Zhang2
• 00DI Theorem Theorem 1.1 .
• 00DJ Proposition Proposition 3.1 .
• 00DK Proof Proof.
• 00DL Proof Proof of the diameter lower bound in Theorem 1.1 .
• 00DM Proof Proof of the diameter upper bound in Theorem 1.1 .
• 00DN Section Introduction
• 00DP Section Volume form asymptotics
• 00DQ Section Diameter lower bound
• 00DR Section Diameter upper bound
• 00DS Equation volform
• 00DT Equation defn
• 00DU Equation calcul
• 00DV Equation ref
• 00DW Equation grad
• 00DX Equation part1
• 00DY Equation part2
• 00DZ Equation part3
• 00E0 Equation lb
• 00E1 Equation totalint2
• 00E2 Equation ub
• 00E3 Equation l1
• 00E4 Bibliography Br
• 00E5 Bibliography Bo
• 00E6 Bibliography BJ
• 00E7 Bibliography CC
• 00E8 Bibliography DPS
• 00E9 Bibliography FGS
• 00EA Bibliography Gr
• 00EB Bibliography Large complex structure limits of $K3$ surfaces
• 00EC Bibliography Ka
• 00ED Bibliography KKMS
• 00EE Bibliography KS
• 00EF Bibliography Li
• 00EG Bibliography Li2
• 00EH Bibliography Li3
• 00EI Bibliography KNX
• 00EJ Bibliography MN
• 00EK Bibliography NX
• 00EL Bibliography RZ
• 00EM Bibliography So
• 00EN Bibliography SYZ
• 00EP Bibliography Ta
• 00EQ Bibliography To0
• 00ER Bibliography To
• 00ES Bibliography To2
• 00ET Bibliography Wa
• 00EU Bibliography Ya
• 00EV Bibliography Zh
• 00EW Theorem Theorem 1.1 .
• 00EX Definition Definition 2.1 .
• 00EY Definition Definition 2.2 (Wilson [ W1 ] ) .
• 00EZ Definition Definition 2.3 .
• 00F0 Theorem Theorem 2.1 (Kawamata) .
• 00F1 Theorem Theorem 2.2 (Iitaka) .
• 00F2 Theorem Theorem 2.3 (Kawamata) .
• 00F3 Conjecture Conjecture 2.1 .
• 00F4 Lemma Lemma 3.1 .
• 00F5 Proof Proof.
• 00F6 Lemma Lemma 3.2 .
• 00F7 Proof Proof.
• 00F8 Proposition Proposition 4.1 .
• 00F9 Proof Proof.
• 00FA Proof Proof of Theorem 1.1 .
• 00FB Conjecture Conjecture 6.1 .
• 00FC Section Introduction
• 00FD Section Some facts from algebraic geometry
• 00FE Section Preliminary remarks
• 00FF Section Limits of Ricci-flat metrics
• 00FG Section Examples
• 00FH Section Further directions
• 00FI Equation length1
• 00FJ Equation volume
• 00FK Equation volume2
• 00FL Equation scrittura
• 00FM Equation integral
• 00FN Equation ma
• 00FP Equation max
• 00FQ Bibliography Ricci curvature bounds and Einstein metrics on compact manifolds
• 00FR Bibliography \'Equations du type Monge-Amp\`ere sur les vari\'et\'es k\"ahleriennes compactes
• 00FS Bibliography Bubbling out of Einstein manifolds
• 00FT Bibliography On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth
• 00FU Bibliography Compact complex surfaces
• 00FV Bibliography Vari\'et\'es K\"ahleriennes dont la premi\`ere classe de Chern est nulle
• 00FW Bibliography The Dirichlet problem for a complex Monge-Amp\`ere equation
• 00FX Bibliography K\"ahler-Ricci flow and the Minimal Model Program for projective varieties
• 00FY Bibliography On the singularities of spaces with bounded Ricci curvature
• 00FZ Bibliography Regularization of closed positive currents and intersection theory
• 00G0 Bibliography Monge-Amp\`ere operators, Lelong numbers and intersection theory
• 00G1 Bibliography Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold
• 00G2 Bibliography K\"ahler manifolds with numerically effective Ricci class
• 00G3 Bibliography Compact complex manifolds with numerically effective tangent bundles
• 00G4 Bibliography Singular K\"ahler-Einstein metrics
• 00G5 Bibliography Constant scalar curvature K\"ahler metrics on fibred complex surfaces
• 00G6 Bibliography Log abundance for surfaces
• 00G7 Bibliography Black hole condensation and the unification of string vacua
• 00G8 Bibliography Metric structures for Riemannian and non-Riemannian spaces
• 00G9 Bibliography Large complex structure limits of $K3$ surfaces
• 00GA Bibliography Numerical Ricci-flat metrics on $K3$
• 00GB Bibliography Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces
• 00GC Bibliography On the cone of divisors of Calabi-Yau fiber spaces
• 00GD Bibliography Introduction to the minimal model problem
• 00GE Bibliography Einstein-K\"ahler $V$-metrics on open Satake $V$-surfaces with isolated quotient singularities
• 00GF Bibliography Polarized period map for generalized $K3$ surfaces and the moduli of Einstein metrics
• 00GG Bibliography The complex Monge-Amp\`ere equation
• 00GH Bibliography Positivity in algebraic geometry I \& II
• 00GI Bibliography Deformations of calibrated submanifolds
• 00GJ Bibliography Dynamics on $K3$ surfaces: Salem numbers and Siegel disks
• 00GK Bibliography Geometry of higher-dimensional algebraic varieties
• 00GL Bibliography Stable base loci of linear series
• 00GM Bibliography On the Albanese map of compact K\"ahler manifolds with numerically effective Ricci curvature
• 00GN Bibliography Regularity properties of the degenerate Monge-Amp\`ere equations on compact K\"ahler manifolds
• 00GP Bibliography Geometric transitions
• 00GQ Bibliography On the convergence and collapsing of K\"ahler metrics
• 00GR Bibliography sy
• 00GS Bibliography Every $K3$ surface is K\"ahler
• 00GT Bibliography The K\"ahler-Ricci flow on surfaces of positive Kodaira dimension
• 00GU Bibliography On Calabi's conjecture for complex surfaces with positive first Chern class
• 00GV Bibliography On the K\"ahler-Ricci flow on projective manifolds of general type
• 00GW Bibliography Applications of the K\"ahler-Einstein-Calabi-Yau metric to moduli of $K3$ surfaces
• 00GX Bibliography Existence and degeneration of K\"ahler-Einstein metrics on minimal algebraic varieties of general type
• 00GY Bibliography The K\"ahler cone on Calabi-Yau threefolds
• 00GZ Bibliography Metric limits of Calabi-Yau manifolds
• 00H0 Bibliography Calabi's conjecture and some new results in algebraic geometry
• 00H1 Bibliography On the Ricci curvature of a compact K\"ahler manifold and the complex Monge-Amp\`ere equation, I
• 00H2 Bibliography Problem section
• 00H3 Bibliography Open problems in geometry
• 00H4 Theorem Theorem 1.1 .
• 00H5 Definition Definition 2.1 .
• 00H6 Definition Definition 2.2 .
• 00H7 Definition Definition 2.3 .
• 00H8 Theorem Theorem 2.4 .
• 00H9 Theorem Theorem 2.5 .
• 00HA Corollary Corollary 2.6 .
• 00HB Definition Definition 2.7 .
• 00HC Remark Remark 2.8 .
• 00HD Definition Definition 2.9 .
• 00HE Remark Remark 2.10 .
• 00HF Lemma Lemma 2.11 .
• 00HG Definition Definition 2.12 .
• 00HH Lemma Lemma 2.13 .
• 00HI Proof Proof.
• 00HJ Corollary Corollary 2.14 .
• 00HK Proof Proof.
• 00HL Proposition Proposition 2.15 .
• 00HM Definition Definition 2.16 .
• 00HN Example Example 2.17 .
• 00HP Definition Definition 2.18 .
• 00HQ Definition Definition 2.19 .
• 00HR Remark Remark 2.20 .
• 00HS Proposition Proposition 2.21 .
• 00HT Definition Definition 2.22 .
• 00HU Lemma Lemma 2.23 .
• 00HV Definition Definition 2.24 .
• 00HW Proposition Proposition 2.25 .
• 00HX Proposition Proposition 2.26 .
• 00HY Lemma Lemma 2.27 .
• 00HZ Proof Proof.
• 00I0 Definition Definition 2.28 .
• 00I1 Remark Remark 2.29 .
• 00I2 Proposition Proposition 2.30 .
• 00I3 Definition Definition 2.31 .
• 00I4 Remark Remark 2.32 .
• 00I5 Proposition Proposition 2.33 .
• 00I6 Definition Definition 2.34 .
• 00I7 Proposition Proposition 2.35 .
• 00I8 Definition Definition 2.36 .
• 00I9 Remark Remark 2.37 .
• 00IA Definition Definition 2.38 .
• 00IB Remark Remark 2.39 .
• 00IC Definition Definition 2.40 .
• 00ID Remark Remark 2.41 .
• 00IE Example Example 2.42 .
• 00IF Proposition Proposition 2.43 .
• 00IG Proof Proof.
• 00IH Proposition Proposition 2.44 .
• 00II Proof Proof.
• 00IJ Proposition Proposition 2.45 .
• 00IK Proof Proof.
• 00IL Lemma Lemma 2.46 .
• 00IM Lemma Lemma 2.47 .
• 00IN Corollary Corollary 2.48 .
• 00IP Corollary Corollary 2.49 .
• 00IQ Definition Definition 2.50 .
• 00IR Proposition Proposition 2.51 .
• 00IS Proposition Proposition 2.52 .
• 00IT Corollary Corollary 2.53 .
• 00IU Proposition Proposition 2.54 .
• 00IV Corollary Corollary 2.55 .
• 00IW Proposition Proposition 2.56 .
• 00IX Proposition Proposition 2.57 .
• 00IY Corollary Corollary 2.58 .
• 00IZ Remark Remark 2.59 .
• 00J0 Definition Definition 2.60 .
• 00J1 Lemma Lemma 2.61 .
• 00J2 Example Example 2.62 .
• 00J3 Lemma Lemma 2.63 .
• 00J4 Corollary Corollary 2.64 .
• 00J5 Definition Definition 2.65 .
• 00J6 Definition Definition 2.66 .
• 00J7 Definition Definition 2.67 .
• 00J8 Theorem Theorem 2.68 .
• 00J9 Corollary Corollary 2.69 .
• 00JA Definition Definition 2.70 .
• 00JB Remark Remark 2.71 .
• 00JC Definition Definition 2.72 .
• 00JD Proposition Proposition 2.73 .
• 00JE Definition Definition 2.74 .
• 00JF Definition Definition 2.75 .
• 00JG Lemma Lemma 2.76 .
• 00JH Proposition Proposition 2.77 .
• 00JI Corollary Corollary 2.78 .
• 00JJ Proposition Proposition 2.79 .
• 00JK Lemma Lemma 2.80 .
• 00JL Theorem Theorem 2.81 .
• 00JM Remark Remark 2.82 .
• 00JN Definition Definition 2.83 .
• 00JP Definition Definition 2.84 .
• 00JQ Lemma Lemma 2.85 .
• 00JR Proposition Proposition 2.86 .
• 00JS Proposition Proposition 2.87 .
• 00JT Proposition Proposition 2.88 .
• 00JU Proof Proof.
• 00JV Definition Definition 2.89 .
• 00JW Definition Definition 2.90 .
• 00JX Proposition Proposition 2.91 .
• 00JY Proposition Proposition 2.92 .
• 00JZ Proof Proof.
• 00K0 Definition Definition 2.93 .
• 00K1 Proposition Proposition 2.94 .
• 00K2 Theorem Theorem 2.95 .
• 00K3 Proposition Proposition 3.1 .
• 00K4 Proof Proof.
• 00K5 Corollary Corollary 3.2 .
• 00K6 Proof Proof.
• 00K7 Remark Remark 3.3 .
• 00K8 Lemma Lemma 3.4 .
• 00K9 Proof Proof.
• 00KA Proposition Proposition 3.5 .
• 00KB Proof Proof.
• 00KC Corollary Corollary 3.6 .
• 00KD Proof Proof.
• 00KE Lemma Lemma 3.7 .
• 00KF Proof Proof.
• 00KG Lemma Lemma 3.8 .
• 00KH Lemma Lemma 3.9 .
• 00KI Proof Proof.
• 00KJ Lemma Lemma 3.10 .
• 00KK Proof Proof.
• 00KL Proposition Proposition 3.11 .
• 00KM Proposition Proposition 3.12 .
• 00KN Proof Proof.
• 00KP Lemma Lemma 3.13 .
• 00KQ Proof Proof.
• 00KR Proposition Proposition 3.14 .
• 00KS Proof Proof.
• 00KT Corollary Corollary 3.15 .
• 00KU Proof Proof.
• 00KV Proposition Proposition 3.16 .
• 00KW Proof Proof.
• 00KX Corollary Corollary 3.17 .
• 00KY Proof Proof.
• 00KZ Remark Remark 3.18 .
• 00L0 Lemma Lemma 3.19 .
• 00L1 Proof Proof.
• 00L2 Proposition Proposition 3.20 .
• 00L3 Proof Proof.
• 00L4 Proposition Proposition 3.21 .
• 00L5 Proof Proof.
• 00L6 Corollary Corollary 3.22 .
• 00L7 Proof Proof.
• 00L8 Corollary Corollary 3.23 .
• 00L9 Proof Proof.
• 00LA Proposition Proposition 3.24 .
• 00LB Proof Proof.
• 00LC Lemma Lemma 3.25 .
• 00LD Proof Proof.
• 00LE Proposition Proposition 3.26 .
• 00LF Proof Proof.
• 00LG Corollary Corollary 3.27 .
• 00LH Proof Proof.
• 00LI Theorem Theorem 3.28 .
• 00LJ Proof Proof.
• 00LK Remark Remark 3.29 .
• 00LL Lemma Lemma 4.1 .
• 00LM Proof Proof.
• 00LN Proposition Proposition 4.2 .
• 00LP Proof Proof.
• 00LQ Remark Remark 4.3 .
• 00LR Proposition Proposition 4.4 .
• 00LS Proof Proof.
• 00LT Theorem Theorem 4.5 .
• 00LU Proof Proof.
• 00LV Proposition Proposition 5.1 .
• 00LW Proof Proof.
• 00LX Claim Claim 5.2 .
• 00LY Proof Proof.
• 00LZ Proposition Proposition 5.3 .
• 00M0 Proof Proof.
• 00M1 Corollary Corollary 5.4 .
• 00M2 Lemma Lemma 5.5 .
• 00M3 Proof Proof.
• 00M4 Proposition Proposition 5.6 .
• 00M5 Proof Proof.
• 00M6 Proposition Proposition 5.7 .
• 00M7 Proof Proof.
• 00M8 Corollary Corollary 5.8 .
• 00M9 Proof Proof.
• 00MA Proposition Proposition 5.9 .
• 00MB Proof Proof.
• 00MC Remark Remark 5.10 .
• 00MD Theorem Theorem 5.11 .
• 00ME Proof Proof.
• 00MF Section Introduction
• 00MG Section Reminders on ultrametric functional analysis
• 00MH Subsection Seminormed vector spaces
• 00MI Subsection Basic constructions
• 00MJ Subsection Orthogonal basis
• 00MK Subsection Banach algebra
• 00ML Subsection Basic constructions
• 00MM Subsection Spectrum
• 00MN Subsection Continuous map
• 00MP Subsection Spectral seminorm
• 00MQ Subsection Banach module
• 00MR Subsection Affinoid algebras
• 00MS Subsection Basic constructions
• 00MT Subsection Algebraic structures: Noetherianity
• 00MU Subsection Topological structures: the spectral norm
• 00MV Subsection Affinoid space as locally ringed space
• 00MW Subsection Spectral calculus
• 00MX Subsection Holomorphic envelop
• 00MY Subsection Holomorphic functional calculus
• 00MZ Subsection Analytification of scheme of finite type
• 00N0 Subsection Local situation
• 00N1 Subsection Global situation
• 00N2 Section Normed section algebra
• 00N3 Subsection Basic setting
• 00N4 Subsection Algebraic properties of normed section algebra
• 00N5 Subsection Spectrum of normed section algebra
• 00N6 Subsection Fubini-Study metrics
• 00N7 Subsection Dual unit disc bundle
• 00N8 Subsection Comparison of algebra norms
• 00N9 Section Geometric approximation
• 00NA Subsection Localization of spectrum by affinoid domain covering
• 00NB Subsection Localization of Banach algebra homomorphism
• 00NC Section Algebraic approximation
• 00ND Subsection Algebra norm induced by Fubini-Study metric
• 00NE Subsection Case for ( ℙ d , 𝒪 ⁡ ( 1 ) )
• 00NF Subsection Case for general ( X , L )
• 00NG Subsection Algebra norm induced by asymptotic Fubini-Study metric
• 00NH Equation Equ: uniform upper bound
• 00NI Equation Equ: Tian bound
• 00NJ Equation Equ: Bost-Randriam bound
• 00NK Reference Item: algebra norm
• 00NL Reference Item: FS metric and FS envelop metric
• 00NM Reference Item: metric
• 00NN Reference Item: metric and weight
• 00NP Reference point in disc bundle global
• 00NQ Reference point in disc bundle local
• 00NR Reference point in disc bundle metric
• 00NS Bibliography Th\'eor\`eme de Hilbert-Samuel "arithm\'etique"
• 00NT Bibliography Spectral theory and analytic geometry over non-Archimedean fields.
• 00NU Bibliography Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry
• 00NV Bibliography Singular semipositive metrics in non-Archimedean geometry
• 00NW Bibliography Non-Archimedean analysis
• 00NX Bibliography Arithmetic positivity on toric varieties
• 00NY Bibliography Germs of analytic varieties in algebraic varieties
• 00NZ Bibliography Espaces vectoriels topologiques. {C}hapitres 1 \`a 5
• 00P0 Bibliography Arithmetic geometry of toric varieties.Metrics, measures and heights.
• 00P1 Bibliography Formes diff\'erentielles r\'eelles et courants sur les espaces de Berkovich
• 00P2 Bibliography Extension property of semipositive invertible sheaves over a non-archimedean field
• 00P3 Bibliography EGA
• 00P4 Bibliography Rigid analytic geometry and its applications
• 00P5 Bibliography Local heights of subvarieties over non-{A}rchimedean fields
• 00P6 Bibliography On Zhang's semipositive metrics
• 00P7 Bibliography Theory of Stein spaces
• 00P8 Bibliography Un th\'{e}or\`eme de prolongement {$L^2$} de sections holomorphes d'un fibr\'{e} hermitien
• 00P9 Bibliography Free basis consisting of strictly small sections
• 00PA Bibliography Notions of Stein spaces in non-archimedean geometry
• 00PB Bibliography On the extension of {$L^2$} holomorphic functions
• 00PC Bibliography M\'etriques de sous-quotient et th\'eor\`eme de {H}ilbert-{S}amuel arithm\'etique pour les faisceaux coh\'erents
• 00PD Bibliography Faisceaux alg\'ebriques coh\'erents
• 00PE Bibliography Introduction to {B}erkovich analytic spaces
• 00PF Bibliography On a set of polarized {K}\"ahler metrics on algebraic manifolds
• 00PG Bibliography Positive line bundles on arithmetic varieties
• 00PH Theorem Theorem 1.1 .
• 00PI Acknowledgement Acknowledgement .
• 00PJ Theorem Theorem 2.1 .
• 00PK Remark Remark 2.2 .
• 00PL Remark Remark 2.3 .
• 00PM Theorem Theorem 2.4 .
• 00PN Remark Remark 2.5 .
• 00PP Remark Remark 2.6 .
• 00PQ Theorem Theorem 2.7 .
• 00PR Lemma Lemma 2.8 .
• 00PS Lemma Lemma 2.9 .
• 00PT Proof Proof.
• 00PU Lemma Lemma 2.10 .
• 00PV Proof Proof.
• 00PW Remark Remark 2.11 .
• 00PX Corollary Corollary 2.12 .
• 00PY Proof Proof.
• 00PZ Theorem Theorem 2.13 .
• 00Q0 Theorem Theorem 2.14 .
• 00Q1 Remark Remark 2.15 .
• 00Q2 Remark Remark 2.16 .
• 00Q3 Example Example 3.1 .
• 00Q4 Proposition Proposition 3.2 .
• 00Q5 Proof Proof.
• 00Q6 Example Example 3.3 .
• 00Q7 Lemma Lemma 3.4 .
• 00Q8 Proof Proof.
• 00Q9 Notation Notation .
• 00QA Lemma Lemma 3.5 .
• 00QB Lemma Lemma 3.6 .
• 00QC Proof Proof.
• 00QD Remark Remark 3.7 .
• 00QE Example Example 3.8 .
• 00QF Remark Remark 3.9 .
• 00QG Remark Remark 3.10 .
• 00QH Example Example 3.11 .
• 00QI Proposition Proposition 3.12 .
• 00QJ Proof Proof.
• 00QK Remark Remark 3.13 .
• 00QL Proposition Proposition 3.14 .
• 00QM Proof Proof.
• 00QN Remark Remark 3.15 .
• 00QP Proposition Proposition 3.16 .
• 00QQ Proof Proof.
• 00QR Definition Definition 3.17 .
• 00QS Example Example 3.18 .
• 00QT Proposition Proposition 3.19 .
• 00QU Proof Proof.
• 00QV Remark Remark 3.20 .
• 00QW Remark Remark 3.21 .
• 00QX Definition Definition 3.22 .
• 00QY Remark Remark 3.23 .
• 00QZ Remark Remark 3.24 .
• 00R0 Remark Remark 3.25 .
• 00R1 Proposition Proposition 3.26 .
• 00R2 Proof Proof.
• 00R3 Notation Notation .
• 00R4 Proposition Proposition 3.27 .
• 00R5 Proof Proof.
• 00R6 Corollary Corollary 3.28 .
• 00R7 Proof Proof.
• 00R8 Definition Definition 3.29 .
• 00R9 Remark Remark 3.30 .
• 00RA Notation Notation .
• 00RB Proposition Proposition 4.1 .
• 00RC Proof Proof.
• 00RD Remark Remark 4.2 .
• 00RE Lemma Lemma 4.3 .
• 00RF Proof Proof.
• 00RG Proposition Proposition 4.4 .
• 00RH Proof Proof.
• 00RI Remark Remark 4.5 .
• 00RJ Lemma Lemma 4.6 .
• 00RK Proof Proof.
• 00RL Corollary Corollary 4.7 .
• 00RM Proof Proof.
• 00RN Corollary Corollary 4.8 .
• 00RP Proof Proof.
• 00RQ Remark Remark 4.9 .
• 00RR Lemma Lemma 4.10 .
• 00RS Proof Proof.
• 00RT Corollary Corollary 4.11 .
• 00RU Lemma Lemma 4.12 .
• 00RV Proof Proof.
• 00RW Remark Remark 4.13 .
• 00RX Proposition Proposition 4.14 .
• 00RY Proof Proof.
• 00RZ Corollary Corollary 4.15 .
• 00S0 Proof Proof.
• 00S1 Remark Remark 4.16 .
• 00S2 Notation Notation .
• 00S3 Notation Notation .
• 00S4 Proposition Proposition 4.17 .
• 00S5 Proof Proof.
• 00S6 Lemma Lemma 4.18 .
• 00S7 Proof Proof.
• 00S8 Remark Remark 4.19 .
• 00S9 Corollary Corollary 4.20 .
• 00SA Proof Proof.
• 00SB Theorem Theorem 4.21 .
• 00SC Proof Proof.
• 00SD Remark Remark 4.22 .
• 00SE Theorem Theorem 4.23 .
• 00SF Proof Proof.
• 00SG Theorem Theorem 4.24 .
• 00SH Proof Proof.
• 00SI Remark Remark 4.25 .
• 00SJ Corollary Corollary 4.26 .
• 00SK Theorem Theorem 5.1 .
• 00SL Lemma Lemma 5.2 .
• 00SM Proof Proof.
• 00SN Lemma Lemma 5.3 .
• 00SP Proof Proof.
• 00SQ Proof Proof.
• 00SR Corollary Corollary 5.4 .
• 00SS Remark Remark 5.5 .
• 00ST Theorem Theorem 5.6 .
• 00SU Notation Notation .
• 00SV Remark Remark 5.7 .
• 00SW Corollary Corollary 5.8 .
• 00SX Remark Remark 5.9 .
• 00SY Theorem Theorem 5.10 .
• 00SZ Proposition Proposition 5.11 .
• 00T0 Proof Proof.
• 00T1 Lemma Lemma 5.12 .
• 00T2 Proof Proof.
• 00T3 Theorem Theorem 5.13 .
• 00T4 Remark Remark 5.14 .
• 00T5 Proof Proof.
• 00T6 Section Introduction
• 00T7 Section Analytic backgrounds
• 00T8 Subsection Skoda inequality
• 00T9 Subsection Kolodziej’s estimate on pluripotentials
• 00TA Subsection Algebraic metrics and asymptotes
• 00TB Subsection Extension of Kähler currents
• 00TC Subsection Savin’s small perturbation theorem
• 00TD Subsection Regularity theory for real Monge-Ampère
• 00TE Subsection Special Lagrangian fibration
• 00TF Section Degenerating Calabi-Yau hypersurfaces
• 00TG Subsection Complex structure
• 00TH Subsection Piecewise linear structure
• 00TI Subsection Kählerian polarisation
• 00TJ Subsection Extension property and locally convex functions
• 00TK Subsection Extension property: the Fermat case
• 00TL Section Estimates on the Kähler potential
• 00TM Subsection Harnack inequality
• 00TN Subsection Local potentials: convexity
• 00TP Subsection Local potentials: plurisubharmonicity
• 00TQ Subsection Locally convex function
• 00TR Subsection Legendre transform, extension, regularisation
• 00TS Subsection Improved Skoda inequality
• 00TT Subsection L ∞ and stability estimates for CY potentials
• 00TU Section Fermat case: Metric convergence and SYZ fibration
• 00TV Subsection Limiting real MA metric
• 00TW Subsection Higher regularity in the generic region
• 00TX Subsection Gromov-Hausdorff convergence
• 00TY Subsection Special Lagrangian fibration in the generic region
• 00TZ Equation Skodaassumption
• 00U0 Equation volumecapacity
• 00U1 Equation L2metrictoriccase
• 00U2 Equation FubiniStudypotential
• 00U3 Equation SlagZhang
• 00U4 Equation holomorphicvolumetoricregion
• 00U5 Equation holomorphicvolumetoricboundary
• 00U6 Equation normalisedcanonicalmeasure
• 00U7 Equation exponentialmeasuredecayawayfromtoric
• 00U8 Equation admissible
• 00U9 Equation localpotentialscomplexsetting
• 00UA Equation CalabiYaucondition
• 00UB Equation ainfty
• 00UC Equation hypersurface
• 00UD Equation holomorphicvolume
• 00UE Equation Lipschitzbound
• 00UF Equation Lipschitzboundonu
• 00UG Equation LocalSkodapreperation
• 00UH Equation convergencetorealMAequation
• 00UI Equation CYmetricsemiflat1
• 00UJ Equation CYmetricsemiflat2
• 00UK Bibliography Blocki
• 00UL Bibliography Blockilecture
• 00UM Bibliography Boucksom1
• 00UN Bibliography Boucksom
• 00UP Bibliography Boucksomsurvey
• 00UQ Bibliography Caff1
• 00UR Bibliography Caff2
• 00US Bibliography Caff4
• 00UT Bibliography CaffarelliViaclovsky
• 00UU Bibliography CollinsTosatti
• 00UV Bibliography Coman
• 00UW Bibliography DemaillyPali
• 00UX Bibliography Donaldsontoric
• 00UY Bibliography DonaldsonSun
• 00UZ Bibliography EGZ
• 00V0 Bibliography EGZ2
• 00V1 Bibliography Lorenzo
• 00V2 Bibliography Gross
• 00V3 Bibliography Gross2
• 00V4 Bibliography GrossTosattiZhang
• 00V5 Bibliography GrossWilson
• 00V6 Bibliography GZ
• 00V7 Bibliography HaaseZharkov
• 00V8 Bibliography HaaseZharkov2
• 00V9 Bibliography HarveyLawson
• 00VA Bibliography HSVZ
• 00VB Bibliography Joyce
• 00VC Bibliography KS2
• 00VD Bibliography KS
• 00VE Bibliography Li
• 00VF Bibliography Mooney
• 00VG Bibliography Odaka
• 00VH Bibliography Savin
• 00VI Bibliography SYZ
• 00VJ Bibliography HSVZ2
• 00VK Bibliography Tosattidiameter
• 00VL Bibliography Tosatti
• 00VM Bibliography Yau
• 00VN Bibliography Zharkov
• 00VP Bibliography Zeriahi
• 00VQ Bibliography Zhang
• 00VR Theorem Theorem 1.1 (Theorem 3.1 of [ To2 ] ) .
• 00VS Theorem Theorem 1.2 .
• 00VT Theorem Theorem 2.1 ( [ DP , EGZ2 ] ) .
• 00VU Theorem Theorem 2.2 .
• 00VV Theorem Theorem 2.3 .
• 00VW Theorem Theorem 2.4 .
• 00VX Lemma Lemma 3.1 .
• 00VY Proof Proof.
• 00VZ Lemma Lemma 3.2 .
• 00W0 Proof Proof.
• 00W1 Lemma Lemma 3.3 .
• 00W2 Proof Proof.
• 00W3 Lemma Lemma 3.4 .
• 00W4 Proof Proof of Theorem 2.2 .
• 00W5 Proof Proof of Theorem 2.3 .
• 00W6 Proposition Proposition 4.1 (cfr. [ ST2 ] ) .
• 00W7 Proof Proof.
• 00W8 Theorem Theorem 4.1 .
• 00W9 Proof Proof.
• 00WA Section Introduction
• 00WB Section Complex Monge-Ampère equations
• 00WC Section A priori estimates
• 00WD Section Collapsing of Ricci-flat metrics
• 00WE Section Examples and remarks
• 00WF Equation jacob
• 00WG Equation c2
• 00WH Equation c3
• 00WI Equation sigma
• 00WJ Equation hack3
• 00WK Equation hack
• 00WL Equation degma1
• 00WM Equation normconst
• 00WN Equation degma2
• 00WP Equation linftybound
• 00WQ Equation c2a
• 00WR Equation schw
• 00WS Equation oneway
• 00WT Equation utile
• 00WU Equation stima1
• 00WV Equation calcc1
• 00WW Equation calcc2
• 00WX Equation stima2
• 00WY Equation stima4
• 00WZ Equation comp1
• 00X0 Equation stima6
• 00X1 Equation hack2
• 00X2 Equation stima7
• 00X3 Equation stima8
• 00X4 Equation theother
• 00X5 Equation stima10
• 00X6 Equation oneway2
• 00X7 Equation sobolev
• 00X8 Equation theother2
• 00X9 Equation stima13
• 00XA Equation thethird
• 00XB Equation thefourth
• 00XC Equation stima14
• 00XD Equation stima15
• 00XE Reference stimaa
• 00XF Equation stimab
• 00XG Equation stimac
• 00XH Equation stima18
• 00XI Equation lowerr
• 00XJ Equation volform
• 00XK Equation maineq
• 00XL Equation linft
• 00XM Equation to8
• 00XN Equation to0
• 00XP Equation to9
• 00XQ Equation to3
• 00XR Equation to4
• 00XS Equation wp
• 00XT Equation wp2
• 00XU Equation wp3
• 00XV Equation to5
• 00XW Equation to6
• 00XX Equation to1
• 00XY Equation to2
• 00XZ Bibliography allard
• 00Y0 Bibliography Degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds
• 00Y1 Bibliography Singular K\"ahler-Einstein metrics
• 00Y2 Bibliography A priori $L^\infty$-estimates for degenerate complex Monge-Amp\`ere equations
• 00Y3 Bibliography Constant scalar curvature K\"ahler metrics on fibred complex surfaces
• 00Y4 Bibliography fs
• 00Y5 Bibliography Large complex structure limits of $K3$ surfaces
• 00Y6 Bibliography hebey
• 00Y7 Bibliography The complex Monge-Amp\`ere equation
• 00Y8 Bibliography A uniform $L\sp \infty$ estimate for complex Monge-Amp\`ere equations
• 00Y9 Bibliography Positivity in algebraic geometry I \& II
• 00YA Bibliography LY
• 00YB Bibliography Dynamics on $K3$ surfaces: Salem numbers and Siegel disks
• 00YC Bibliography Sobolev and mean-value inequalities on generalized submanifolds of $R\sp{n}$
• 00YD Bibliography og
• 00YE Bibliography PSS
• 00YF Bibliography The K\"ahler-Ricci flow on surfaces of positive Kodaira dimension
• 00YG Bibliography SoT2
• 00YH Bibliography stoppa
• 00YI Bibliography topping
• 00YJ Bibliography A general Schwarz Lemma for almost-Hermitian manifolds
• 00YK Bibliography Limits of Calabi-Yau metrics when the K\"ahler class degenerates
• 00YL Bibliography tesi
• 00YM Bibliography Taming symplectic forms and the Calabi-Yau equation
• 00YN Bibliography Metric limits of Calabi-Yau manifolds
• 00YP Bibliography On the Ricci curvature of a compact K\"ahler manifold and the complex Monge-Amp\`ere equation, I
• 00YQ Bibliography A general Schwarz lemma for K\"ahler manifolds
• 00YR Bibliography Problem section
• 00YS Bibliography Degenerate Monge-Amp\`ere equations over projective manifolds
• 00YT Chapter Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori
• 00YU Bibliography V. Batyrev. Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori. Duke Math. J., 69(2):349-409, 1993.
• 00YV Chapter The real Monge-Ampere equation and affine flat structures
• 00YW Bibliography S.-Y. Cheng and S.-T. Yau. The real Monge-Ampere equation and affine flat structures. In Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1 (Beijing, 1980), pages 339-370, Beijing, 1982. Science Press.
• 00YX Chapter Introduction to Toric Varieties
• 00YY Bibliography W. Fulton. Introduction to Toric Varieties, volume 131 of Annals of Math. Studies. Princeton University Press, 1993.
• 00YZ Chapter Gravitational instantons
• 00Z0 Bibliography G. W. Gibbons and S. W. Hawking. Gravitational instantons. Phys. Lett. B, 78:430-432, 1978.
• 00Z1 Chapter Discriminants, Resultants, and Multidimensional Determinants
• 00Z2 Bibliography I. Gelfand, M. Kapranov, and A. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhauser, 1994.
• 00Z3 Bibliography Mark Gross. Topological mirror symmetry. Invent. Math., 144(1):75-137, 2001.
• 00Z4 Chapter Affine manifolds, log structures, and mirror symmetry
• 00Z5 Bibliography M. Gross and B. Siebert. Affine manifolds, log structures, and mirror symmetry. Preprint math.AG/0211094, 2002.
• 00Z6 Chapter Elliptic partial differential equations of second order
• 00Z7 Bibliography D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1983.
• 00Z8 Bibliography M. Gross and P. M. H. Wilson. Large complex structure limits of K3 surfaces. J. Differential Geom., 55(3):475-546, 2000.
• 00Z9 Chapter Gravitational instantons
• 00ZA Bibliography S. W. Hawking. Gravitational instantons. Phys. Lett. A, 60(2):81-83, 1977.
• 00ZB Chapter The moduli space of special Lagrangian submanifolds
• 00ZC Bibliography N. Hitchin. The moduli space of special Lagrangian submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25(3-4):503-515 (1998), 1997. Dedicated to Ennio De Giorgi.
• 00ZD Bibliography C. Haase and I. Zharkov. Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I. Preprint math.AG/0205321, 2002.
• 00ZE Bibliography C. Haase and I. Zharkov. Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces II. Preprint math.AG/0301222, 2003.
• 00ZF Bibliography M. Kontsevich and Y. Soibelman. Homological mirror symmetry and torus fibrations. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 203-263. World Sci. Publishing, River Edge, NJ, 2001.
• 00ZG Bibliography M. Kontsevich and Yu. Tschinkel. Non-Archimedean Kahler geometry. In preparation, 2002.
• 00ZH Chapter Complete Ricci-flat Kahler metrics on C n need not be flat
• 00ZI Bibliography C. LeBrun. Complete Ricci-flat Kahler metrics on C n need not be flat. In Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), volume 52 of Proc. Sympos. Pure Math., pages 297-304. Amer. Math. Soc., Providence, RI, 1991.
• 00ZJ Chapter Mirror symmetry without corrections
• 00ZK Bibliography N. C. Leung. Mirror symmetry without corrections. Preprint math.DG/0009235, 2000.
• 00ZL Chapter From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform
• 00ZM Bibliography N. C. Leung, S. T. Yau, and E. Zaslow. From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform. In Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), volume 23 of AMS/IP Stud. Adv. Math., pages 209-225. Amer. Math. Soc., Providence, RI, 2001.
• 00ZN Chapter Constructions of Calabi-Yau metrics and of special Lagrangian submanifolds
• 00ZP Bibliography D. Matessi. Constructions of Calabi-Yau metrics and of special Lagrangian submanifolds. Ph.D Thesis, University of Warwick, UK, 2001.
• 00ZQ Chapter Amoebas of algebraic varieties
• 00ZR Bibliography G. Mikhalkin. Amoebas of algebraic varieties. Preprint math.AG/0108225, 2001.
• 00ZS Chapter Summing up Dirichlet instantons
• 00ZT Bibliography H. Ooguri and C. Vafa. Summing up Dirichlet instantons. Phys. Rev. Lett., 77(16):3296-3298, 1996.
• 00ZU Chapter Hamiltonian constructions of Kahler-Einstein metrics and Kahler metrics of constant scalar curvature
• 00ZV Bibliography H. Pedersen and Y. S. Poon. Hamiltonian constructions of Kahler-Einstein metrics and Kahler metrics of constant scalar curvature. Comm. Math. Phys., 136(2):309-326, 1991.
• 00ZW Bibliography A. Strominger, S.-T. Yau, and E. Zaslow. Mirror symmetry is T-duality. Nuclear Phys. B, 479(1-2):243-259, 1996.
• 00ZX Chapter Projective capacity
• 00ZY Bibliography H. A LEXANDER, Projective capacity, Ann. of Math. Studies, Conference on Several Complex Variables 100(1981), Princeton, pp. 3-27.
• 00ZZ Chapter Comparison of two capacities in Cⁿ
• 0100 Bibliography H. A LEXANDER & B.A. TAYLOR, Comparison of two capacities in Cn , Math. Zeit. 186(1984), 407-417.
• 0101 Chapter A new capacity for plurisubharmonic functions
• 0102 Bibliography E. B EDFORD & B.A. TAYLOR, A new capacity for plurisubharmonic functions, Acta Math. 149(1982), 1-40.
• 0103 Chapter Plurisubharmonic functions with logarithmic singularities
• 0104 Bibliography , Plurisubharmonic functions with logarithmic singularities, Ann. Inst. Fourier, Grenoble 38(1998), 133-171.
• 0105 Chapter Estimates for the complex Monge-Ampère operator
• 0106 Bibliography Z. B LOCKI, Estimates for the complex Monge-Ampère operator, Bull. Polish Acad. Sci. Math. 41(1993), 151-157.
• 0107 Chapter Local estimates for plurisubharmonic functions
• 0108 Bibliography A. B RUDNYI, Local estimates for plurisubharmonic functions, Annals of Math. 149(1999), 511-533.
• 0109 Chapter Pluricomplex energy
• 010A Bibliography U. C EGRELL, Pluricomplex energy, Acta Math. 180(1998), 187-217.
• 010B Chapter Approximation of plurisubharmonic functions and the general definition of the complex Monge-Ampère operator
• 010C Bibliography , Approximation of plurisubharmonic functions and the general definition of the com- plex Monge-Ampère operator, preprint, Umea University, 1999.
• 010D Chapter Complex Analytic Sets
• 010E Bibliography E.M. C HIRCA, Complex Analytic Sets, Mathematics and its Applications, Kluwer Aca- demic Publishers, 1989.
• 010F Chapter On the size of lemniscates of polynomials in one and several variables
• 010G Bibliography A. C UYT, K. D RIVER & D.S. L UBINSKY , On the size of lemniscates of polynomials in one and several variables, Proc. Amer. Math. Soc. 124-7(1996), 2123-2136.
• 010H Chapter Mesures de Monge-Ampère et caractérisation des variétés algébriques affines
• 010I Bibliography J.P. D EMAILLY, Mesures de Monge-Ampère et caractérisation des variétés algébriques affines, Mémoire de la SMF, nouvelle série 19(1985), 1-124.
• 010J Chapter Mesures de Monge-Ampère et mesures pluriharmoniques
• 010K Bibliography , Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Zeit. 194(1987), 519-564.
• 010L Chapter Potential theory in several complex variables
• 010M Bibliography , Potential theory in several complex variables, Course at the ICPAM summer school on Complex Analysis, Nice, France, 1989.
• 010N Chapter Monge-Ampère operators, Lelong numbers and intersection theory
• 010P Bibliography , Monge-Ampère operators, Lelong numbers and intersection theory, in: Complex Analysis and Geometry, Univ. Series in Math., V. Ancona and A. Silva (eds.), New York, Plenum, 1993.
• 010Q Chapter Note on pull-back and Lelong number of currents
• 010R Bibliography C. FAVRE, Note on pull-back and Lelong number of currents, Bull. Soc. Math. de France, to appear.
• 010S Chapter Dynamique des applications rationnelles des espaces multiprojectifs
• 010T Bibliography C. FAVRE & V. G UEDJ, Dynamique des applications rationnelles des espaces multiprojec- tifs, Prépublication de l’Université de Paris-Sud, 99-41, 1999.
• 010U Chapter An Introduction to Complex Analysis in Several Variables
• 010V Bibliography L. H ÖRMANDER, An Introduction to Complex Analysis in Several Variables, North Hol- land, Amsterdam, 3rd edition, 1990.
• 010W Chapter Notions of Convexity
• 010X Bibliography , Notions of Convexity, Progress in Mathematics, Birkhäuser, Boston, 1994.
• 010Y Chapter Densité des fonctions plurisousharmoniques
• 010Z Bibliography C.O. K ISELMAN, Densité des fonctions plurisousharmoniques, Bull. Soc. Math. de France, 107(1979), 295-304.
• 0110 Chapter Attenuating the singularities of plurisubharmonic functions
• 0111 Bibliography , Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math. 60(1994), 173-197.
• 0112 Chapter Ensembles de sous-niveau et images inverses des fonctions plurisousharmoniques
• 0113 Bibliography , Ensembles de sous-niveau et images inverses des fonctions plurisousharmoniques, Bull. Sci. Math. 2e série, to appear.
• 0114 Chapter Pluripotential Theory
• 0115 Bibliography M. K LIMEK, Pluripotential Theory, Oxford University Press, London, 1991.
• 0116 Bibliography S. KOLODZIEJ, The complex Monge-Ampère equation, Acta Math. 180(1998), 69-117.
• 0117 Chapter Fonctions Plurisousharmoniques et Formes Différentielles Positives
• 0118 Bibliography P. L ELONG, Fonctions Plurisousharmoniques et Formes Différentielle Positives, Gordon et Breach, New-York and Dunod, Paris, 1969.
• 0119 Chapter Characterization of quasi-analytic functions of several variables by means of rational approximation
• 011A Bibliography W. P LESNIAK, Characterization of quasi-analytic functions of several variables by means of rational approximation, Ann. Polon. Math. 27(1973), 149-157.
• 011B Chapter Function Theory in the Unit Ball of Cⁿ
• 011C Bibliography W. R UDIN, Function Theory in the Unit Ball of Cn , Springer Verlag, New York, Heidel- gerg, Berlin, 1980.
• 011D Chapter An estimate for polynomials on analytic sets
• 011E Bibliography A. S ADULLAEV, An estimate for polynomials on analytic sets, Math. USSR Izvestiya 20(1983), 493-502.
• 011F Chapter Extremal plurisubharmonic functions in Cᴺ
• 011G Bibliography J. S ICIAK, Extremal plurisubharmonic functions in CN , Ann. Polon. Math. 39(1981), 175-211.
• 011H Chapter Extremal plurisubharmonic functions and capacities in Cᴺ
• 011I Bibliography , Extremal plurisubharmonic functions and capacities in CN , Sophia Kokyuroku in Math. Volume 14, Tokyo, 1982.
• 011J Chapter Sous-ensembles analytiques d’ordre fini ou infini dans Cⁿ
• 011K Bibliography H. S KODA, Sous-ensembles analytiques d’ordre fini ou infini dans Cn , Bull. Soc. Math. de France 100(1972), 353-408.
• 011L Chapter Analyticity of sets associated to Lelong numbers and the extension of closed positive currents
• 011M Bibliography Y.T. S IU, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27(1974), 53-156.
• 011N Chapter Potential Theory in Modern Function Theory
• 011P Bibliography M. T SUJI, Potential Theory in Modern Function Theory, Chelsea, New York, 1975.
• 011Q Chapter Extremal plurisubharmonic functions, orthogonal polynomials and Bernstein-Walsh theorem for analytic functions of several complex variables
• 011R Bibliography V.P. Z AHARIUTA, Extremal plurisubharmonic functions, orthogonal polynomials and Bernstein-Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math. 33(1976), 137-148.
• 011S Chapter Fonction de Green pluricomplexe à pôle à l’infini sur un espace de Stein parabolique
• 011T Bibliography A. Z ERIAHI, Fonction de Green pluricomplexe à pôle à l’infini sur un espace de Stein parabolique, Math. Scand. 69(1991), 89-126.
• 011U Chapter Pluricomplex Green functions and the Dirichlet problem for the complex Monge-Ampère operator
• 011V Bibliography , Pluricomplex Green functions and the Dirichlet problem for the complex Monge- Ampère operator, Michigan Math. J. 44(1997), 579-596.
• 011W Chapter A criterion of algebraicity for Lelong classes and analytic sets
• 011X Bibliography , A criterion of algebraicity for Lelong classes and analytic sets, Acta Math. 184(2000), 113-143.
• 011Y Chapter Ricci curvature bounds and Einstein metrics on compact manifolds
• 011Z Bibliography M.T.Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc. 2 (1989), 455-490.
• 0120 Chapter Convergence and rigidity of manifolds under Ricci curvature bounds
• 0121 Bibliography M.T.Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), 429-445.
• 0122 Chapter The L^2 structure of moduli spaces of Einstein metrics on 4-manifolds
• 0123 Bibliography M.T.Anderson, The L^2 structure of moduli spaces of Einstein metrics on 4-manifolds, G.A.F.A. (1991), 231-251.
• 0124 Chapter Nilpotent structures and invariant metrics on collapsed manifolds
• 0125 Bibliography J.Cheeger, K.Fukaya, M.Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, Joural of the American Mathematical Society, 5 (1992), 327-372.
• 0126 Chapter Collapsing Riemannian manifolds while keeping their curvature bound I
• 0127 Bibliography J.Cheeger, M.Gromov, Collapsing Riemannian manifolds while keeping their curvature bound I, J. Differ. Geom. 23 (1986), 309-364.
• 0128 Chapter Collapsing Riemannian Manifolds while keeping their curvature bounded II
• 0129 Bibliography J.Cheeger, M.Gromov, Collapsing Riemannian Manifolds while keeping their curvature bounded II, J.Diff.Geom. 32 (1990), 269-298.
• 012A Chapter Anti-self-duality of curvature and degeneration of metrics with special holonomy
• 012B Bibliography J.Cheeger, G.Tian, Anti-self-duality of curvature and degeneration of metrics with special holonomy, Commun. Math. Phys. 255 (2005), 391-417.
• 012C Chapter Curvature and injectivity radius estimates for Einstein 4-manifolds
• 012D Bibliography J.Cheeger, G.Tian, Curvature and injectivity radius estimates for Einstein 4-manifolds, Journal of the American Mathematical Society, 19 (2006), 487-525.
• 012E Chapter Degeneration of Einstein metrics and metrics with special holonomy
• 012F Bibliography J.Cheeger, Degeneration of Einstein metrics and metrics with special holonomy, in Surveys in differential geometry VIII, 29-73.
• 012G Chapter On global action-angle coordinates
• 012H Bibliography J.Duistermaat, On global action-angle coordinates, Comm. Pure Appled Math., 33 (1980), 687-706.
• 012I Chapter Hausdorff convergence of Riemannian manifolds and its application
• 012J Bibliography K.Fukaya, Hausdorff convergence of Riemannian manifolds and its application, Advance Studies in Pure Mathematics, 18 (1990), 143-234.
• 012K Chapter Multivalued Morse theory, asymptortic analysis and mirror aymmetry
• 012L Bibliography K.Fukaya, Multivalued Morse theory, asymptortic analysis and mirror aymmetry, Proceedings of Symposia in Pure Mathematics, 73 (2005), 205-278.
• 012M Chapter Lipschitz converges of Riemannian manifolds
• 012N Bibliography R.E.Green, H.Wu, Lipschitz converges of Riemannian manifolds, Pacific J. Math. 131 (1988), 119-141.
• 012P Chapter Metric structures for Riemannian and non-Riemannian spaces
• 012Q Bibliography M.Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhauser 1999.
• 012R Chapter A construction of new families of minmal lagrangian submanifolds via torus action
• 012S Bibliography E.Goldstein, A construction of new families of minmal lagrangian submanifolds via torus action, J. Diff. Geom. 58 (2001), 233-261.
• 012T Chapter Calibrated fibrations
• 012U Bibliography E.Goldstein, Calibrated fibrations, Comm. Anal. Geom. 10 (2002), 127-150.
• 012V Chapter Examples of special lagrangian fibrations
• 012W Bibliography M.Gross, Examples of special lagrangian fibrations, in Symplectic Geometry and Mirror Symmetry, World Scientific Singapore, (2001), 81-109.
• 012X Chapter Special lagrangian fibrations II-Geometry
• 012Y Bibliography M.Gross, Special lagrangian fibrations II-Geometry, Surveys in Differential Geometry: Differential geometry inspired by string theory, International Press, (1999), 341-404.
• 012Z Bibliography M.Gross, P.M.H.Wilson, Large complex structure limits of K3 surfaces, J. Diff. Geom. 55 (2000), 475-546.
• 0130 Bibliography D.Gilbarg, N.S.Trudinger, Elliptic partial differential equations of second two, Springer 1983.
• 0131 Bibliography R.Harvey, H.B.Lawson, Calibrated geometries, Acta Math., 148 (1982), 47-157.
• 0132 Bibliography N.Hitchin, The moduli space of special lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), 503-515.
• 0133 Chapter Compact manifolds with special holonomy
• 0134 Bibliography D.D.Joyce, Compact manifolds with special holonomy, Oxford University Press, 2000.
• 0135 Bibliography D.D.Joyce, Singularities of special lagrangian fibrations and the SYZ conjecture, Comm.Anal.Geom. 11 (2003), 859-907.
• 0136 Chapter Lectures on Calabi-Yau and special Lagrangian geometry
• 0137 Bibliography D.D.Joyce, Lectures on Calabi-Yau and special Lagrangian geometry, math.DG/0108088.
• 0138 Bibliography M.Kontsevich, Y.Soibelman, Homological mirror symmetry and torus fibrations , in Symplectic geometry and mirror symmetry, World Sci. Publishing, (2001), 203-263.
• 0139 Chapter Kahler-Einstein metrics on Kummer threefold and special lagrangian tori
• 013A Bibliography P.Lu, Kahler-Einstein metrics on Kummer threefold and special lagrangian tori, Comm. Anal. Geom. 7 (1999), 787-806.
• 013B Chapter Deformation of calibrated submanifolds
• 013C Bibliography R.C.Mclean, Deformation of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705-747.
• 013D Chapter Riemannian Geometry
• 013E Bibliography P.Petersen, Riemannian Geometry, Springer, 1997.
• 013F Chapter On the convergence and collapsing of Kahler metrics
• 013G Bibliography W.D.Ruan, On the convergence and collapsing of Kahler metrics, J. Differ. Geom. 52 (1999), 1-40.
• 013H Chapter Generalized special Lagrangian torus fibration for Calabi-Yau hypersurfaces in toric varieties. I
• 013I Bibliography W.D.Ruan, Generalized special Lagrangian torus fibration for Calabi-Yau hypersurfaces in toric varieties. I. Commun. Contemp. Math. 9 (2007), no. 2, 201-216.
• 013J Chapter Generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties. II
• 013K Bibliography W.D.Ruan, Generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties. II. in Mirror symmetry. V, Amer. Math. Soc., Providence, RI, 2006. 457-477.
• 013L Chapter Generalized special Lagrangian fibration for Calabi-Yau hypersurfaces in toric varieties III: The smooth fibres
• 013M Bibliography W.D.Ruan, Generalized special Lagrangian fibration for Calabi-Yau hypersurfaces in toric varieties III: The smooth fibres, arXiv:math/0309450.
• 013N Chapter H-minimal Lagrangian fibrations in Kahler manifolds and minimal Lagrangian vanishing tori in Kahler-Einstein manifolds
• 013P Bibliography W.D.Ruan, H-minimal Lagrangian fibrations in Kahler manifolds and minimal Lagrangian vanishing tori in Kahler-Einstein manifolds, arXiv:math/0309177 .
• 013Q Chapter Convergence of Calabi-Yau manifolds
• 013R Bibliography W.D.Ruan, Y.G.Zhang, Convergence of Calabi-Yau manifolds, arXiv:0905.3424.
• 013S Chapter Deformations of special lagrangian submanifolds
• 013T Bibliography S.Salur, Deformations of special lagrangian submanifolds, Commu. Cont. Math. Vol.2, 3 (2000), 365-372.
• 013U Bibliography A.Strominger, S.T.Yau, E.Zaslow, Mirror symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259.
• 013V Bibliography V.Tosatti, Limits of Calabi-Yau metrics when the Kahler class degenerates, arXiv:0710.4571, to appear in J.Eur.Math.Soc. 2009.
• 013W Bibliography R.P.Thomas, S.T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 no. 5 (2002), 1075-1113.
• 013X Bibliography S.T.Yau, On the Ricci curvature of a compact Kahler manifold and complex Monge-Ampere equation I, Comm. Pure Appl. Math. 31 (1978), 339-411.
• 013Y Chapter Einstein manifolds with zero Ricci curvature
• 013Z Bibliography S.T.Yau, Einstein manifolds with zero Ricci curvature, in Lectures on Einstein manifolds, International Press, (1999), 1-14.
• 0140 Theorem Theorem 0.1
• 0141 Theorem Theorem 0.2
• 0142 Definition Definition 1.1
• 0143 Definition Definition 1.2
• 0144 Theorem Theorem 1.3
• 0145 Theorem Theorem 1.4
• 0146 Definition Definition 1.6
• 0147 Theorem Theorem 1.7
• 0148 Definition Definition 1.8
• 0149 Definition Definition 1.9
• 014A Theorem Theorem 1.10
• 014B Theorem Theorem 1.11
• 014C Theorem Theorem 2.1
• 014D Corollary Corollary 2.2
• 014E Theorem Theorem 2.6
• 014F Proposition Proposition 2.7
• 014G Proposition Proposition 2.8
• 014H Definition Definition 2.9
• 014I Theorem Theorem 2.12
• 014J Proposition Proposition 3.3
• 014K Theorem Theorem 4.1
• 014L Proposition Proposition 4.2
• 014M Lemma Lemma 4.3
• 014N Theorem Theorem 4.4
• 014P Section Introduction
• 014Q Theorem Theorem A .
• 014R Corollary Corollary B .
• 014S Theorem Theorem C .
• 014T Section 1. Preliminaries
• 014U Subsection 1.1. Metrics
• 014V Subsection 1.2. Measures and forms
• 014W Lemma Lemma 1.1 .
• 014X Proof Proof.
• 014Y Subsection 1.3. Integral piecewise affine spaces
• 014Z Lemma Lemma 1.2 .
• 0150 Proof Proof.
• 0151 Remark Remark 1.3 .
• 0152 Subsection 1.4. Tropicalizations and polar coordinates
• 0153 Section 2. The hybrid space associated to an snc model
• 0154 Subsection 2.1. The dual complex
• 0155 Subsection 2.2. The hybrid topology
• 0156 Proposition Proposition 2.1 .
• 0157 Lemma Lemma 2.2 .
• 0158 Proof Proof of Proposition 2.1 .
• 0159 Definition Definition 2.3 .
• 015A Example Example 2.4 .
• 015B Section 3. Proof of Theorem A
• 015C Subsection 3.1. Residual measures
• 015D Definition Definition 3.1 .
• 015E Lemma Lemma 3.2 .
• 015F Proof Proof.
• 015G Definition Definition 3.3 .
• 015H Subsection 3.2. Statement and first reductions
• 015I Theorem Theorem 3.4 .
• 015J Lemma Lemma 3.5 .
• 015K Lemma Lemma 3.6 .
• 015L Proof Proof of Lemma 3.6 .
• 015M Subsection 3.3. Proof of Lemma 3.5
• 015N Lemma Lemma 3.7 .
• 015P Lemma Lemma 3.8 .
• 015Q Proof Proof.
• 015R Section 4. The limit hybrid model
• 015S Subsection 4.1. Snc models and simple blowups
• 015T Lemma Lemma 4.1 .
• 015U Proof Proof.
• 015V Subsection 4.2. Induced maps between dual complexes
• 015W Definition Definition 4.2 .
• 015X Proposition Proposition 4.3 .
• 015Y Corollary Corollary 4.4 .
• 015Z Lemma Lemma 4.5 .
• 0160 Proof Proof.
• 0161 Lemma Lemma 4.6 .
• 0162 Proof Proof.
• 0163 Lemma Lemma 4.7 .
• 0164 Proof Proof.
• 0165 Proof Proof of Proposition 4.3 .
• 0166 Subsection 4.3. Induced maps between hybrid spaces
• 0167 Proposition Proposition 4.8 .
• 0168 Proof Proof.
• 0169 Subsection 4.4. The limit hybrid space
• 016A Definition Definition 4.9 .
• 016B Remark Remark 4.10 .
• 016C Subsection 4.5. Convergence of measures
• 016D Corollary Corollary 4.11 .
• 016E Subsection 4.6. The projective case
• 016F Proposition Proposition 4.12 .
• 016G Proof Proof.
• 016H Section 5. Berkovich spaces and skeleta
• 016I Subsection 5.1. Models
• 016J Subsection 5.2. Model metrics
• 016K Subsection 5.3. Log canonical divisors
• 016L Example Example 5.1 .
• 016M Remark Remark 5.2 .
• 016N Subsection 5.4. Log discrepancies
• 016P Example Example 5.3 .
• 016Q Example Example 5.4 .
• 016R Example Example 5.5 .
• 016S Subsection 5.5. The skeleton of a dlt model
• 016T Subsection 5.6. From log discrepancies to Temkin’s metric
• 016U Proposition Proposition 5.6 .
• 016V Lemma Lemma 5.7 .
• 016W Proof Proof.
• 016X Proof Proof of Proposition 5.6 .
• 016Y Subsection 5.7. The skeleton of a metric on K X
• 016Z Definition Definition 5.8 .
• 0170 Definition Definition 5.9 .
• 0171 Proposition Proposition 5.10 .
• 0172 Proof Proof.
• 0173 Subsection 5.8. Residual boundaries
• 0174 Example Example 5.11 .
• 0175 Lemma Lemma 5.12 .
• 0176 Proof Proof.
• 0177 Subsection 5.9. Skeleta and base change
• 0178 Lemma Lemma 5.13 .
• 0179 Proof Proof.
• 017A Lemma Lemma 5.14 .
• 017B Proof Proof.
• 017C Section 6. Skeletal measures
• 017D Subsection 6.1. Residually metrized models
• 017E Definition Definition 6.1 .
• 017F Example Example 6.2 .
• 017G Subsection 6.2. Residual measures
• 017H Definition Definition 6.3 .
• 017I Subsection 6.3. Measures on dual complexes
• 017J Definition Definition 6.4 .
• 017K Subsection 6.4. Skeletal mesures on Berkovich spaces
• 017L Definition Definition 6.5 .
• 017M Lemma Lemma 6.6 .
• 017N Proof Proof.
• 017P Subsection 6.5. Behavior under base change
• 017Q Theorem Theorem 6.7 .
• 017R Proof Proof.
• 017S Lemma Lemma 6.8 .
• 017T Proof Proof of Lemma 6.8 .
• 017U Section 7. The Calabi–Yau case
• 017V Subsection 7.1. Topology of the skeleton
• 017W Subsection 7.2. The skeletal measure
• 017X Theorem Theorem 7.1 .
• 017Y Proof Proof.
• 017Z Remark Remark 7.2 .
• 0180 Section 8. Extensions
• 0181 Subsection 8.1. A singular version of Theorem A
• 0182 Theorem Theorem 8.1 .
• 0183 Corollary Corollary 8.2 .
• 0184 Corollary Corollary 8.3 .
• 0185 Subsection 8.2. Corollary B for pairs
• 0186 Theorem Theorem 8.4 .
• 0187 Proof Proof of Theorem 8.4 .
• 0188 Subsection 8.3. Degenerations of Ricci-flat Kähler manifolds
• 0189 Theorem Theorem 8.5 .
• 018A Proof Proof.
• 018B Section Appendix A Berkovich spaces over Banach rings
• 018C Subsection A.1. Berkovich spectra
• 018D Example Example A.1 .
• 018E Example Example A.2 .
• 018F Subsection A.2. Analytification of a scheme
• 018G Example Example A.3 .
• 018H Subsection A.3. The hybrid norm on ℂ
• 018I Subsection A.4. Hybrid geometry over ℂ
• 018J Subsection A.5. The hybrid circle
• 018K Proposition Proposition A.4 .
• 018L Proof Proof.
• 018M Remark Remark A.5 .
• 018N Subsection A.6. Geometry over the hybrid circle
• 018P Lemma Lemma A.6 .
• 018Q Section 1. Introduction
• 018R Theorem Theorem A .
• 018S Theorem Theorem A’ .
• 018T Acknowledgement Acknowledgment .
• 018U Section 2. Background
• 018V Subsection 2.1. Berkovich space and models
• 018W Subsection 2.2. Model functions
• 018X Definition Definition 2.1 .
• 018Y Proposition Proposition 2.2 .
• 018Z Subsection 2.3. Forms and de Rham classes
• 0190 Definition Definition 2.3 .
• 0191 Definition Definition 2.4 .
• 0192 Subsection 2.4. θ -psh functions
• 0193 Definition Definition 2.5 .
• 0194 Proposition Proposition 2.6 .
• 0195 Proposition Proposition 2.7 .
• 0196 Proposition Proposition 2.8 .
• 0197 Theorem Theorem 2.9 .
• 0198 Theorem Theorem 2.10 .
• 0199 Theorem Theorem 2.11 .
• 019A Corollary Corollary 2.12 .
• 019B Subsection 2.5. Envelopes
• 019C Proposition Proposition 2.13 .
• 019D Definition Definition 2.14 .
• 019E Proposition Proposition 2.15 .
• 019F Subsection 2.6. Metrized line bundles and curvature forms
• 019G Definition Definition 2.16 .
• 019H Theorem Theorem 2.17 .
• 019I Subsection 2.7. Intersection numbers and Monge-Ampère measures
• 019J Proposition Proposition-Definition 2.18 .
• 019K Proof Proof.
• 019L Proposition Proposition 2.19 .
• 019M Proof Proof.
• 019N Proposition Proposition 2.20 .
• 019P Proof Proof.
• 019Q Proposition Proposition 2.21 .
• 019R Proof Proof of Proposition 2.21 .
• 019S Remark Remark 2.22 .
• 019T Subsection 2.8. Radon measures and convergence results
• 019U Lemma Lemma 2.23 .
• 019V Lemma Lemma 2.24 .
• 019W Corollary Corollary 2.25 .
• 019X Proof Proof.
• 019Y Section 3. Monge-Ampère operator on bounded functions
• 019Z Theorem Theorem 3.1 .
• 01A0 Remark Remark 3.2 .
• 01A1 Corollary Corollary 3.3 .
• 01A2 Subsection 3.1. Proof of Theorem 3.1
• 01A3 SourceObject Assertion A(p) .
• 01A4 Lemma Lemma 3.4 .
• 01A5 Proof Proof of Lemma 3.4 .
• 01A6 Lemma Lemma 3.5 .
• 01A7 Proof Proof of Lemma 3.5 .
• 01A8 Definition Definition 3.6 .
• 01A9 Proposition Proposition 3.7 .
• 01AA Proof Proof.
• 01AB Subsection 3.2. The Chambert-Loir measure
• 01AC Section 4. Capacity and quasicontinuity
• 01AD Definition Definition 4.1 .
• 01AE Lemma Lemma 4.2 .
• 01AF Proof Proof.
• 01AG Proposition Proposition 4.3 .
• 01AH Definition Definition 4.4 .
• 01AI Proposition Proposition 4.5 .
• 01AJ Lemma Lemma 4.6 .
• 01AK Proof Proof.
• 01AL Lemma Lemma 4.7 .
• 01AM Proof Proof.
• 01AN Proof Proof of Proposition 4.3 .
• 01AP Proof Proof of Proposition 4.5 .
• 01AQ Section 5. Locality and the comparison principle
• 01AR Theorem Theorem 5.1 .
• 01AS Corollary Corollary 5.2 .
• 01AT Proof Proof.
• 01AU Corollary Corollary 5.3 .
• 01AV Proof Proof.
• 01AW Lemma Lemma 5.4 .
• 01AX Proof Proof.
• 01AY Proof Proof of Theorem 5.1 .
• 01AZ Section 6. Energy
• 01B0 Subsection 6.1. Energy of model functions
• 01B1 Proposition Proposition 6.1 .
• 01B2 Proof Proof.
• 01B3 Subsection 6.2. Energy of ω -psh functions
• 01B4 Proposition Proposition 6.2 .
• 01B5 Proof Proof.
• 01B6 Proposition Proposition 6.3 .
• 01B7 Subsection 6.3. Non-pluripolar Monge-Ampère measures
• 01B8 Definition Definition 6.4 .
• 01B9 Remark Remark 6.5 .
• 01BA Definition Definition 6.6 .
• 01BB Lemma Lemma 6.7 .
• 01BC Proof Proof.
• 01BD Lemma Lemma 6.8 .
• 01BE Proof Proof.
• 01BF Proposition Proposition 6.9 .
• 01BG Proof Proof.
• 01BH Lemma Lemma 6.10 .
• 01BI Proof Proof.
• 01BJ Subsection 6.4. Locality and the comparison principle
• 01BK Proposition Proposition 6.11 .
• 01BL Proof Proof.
• 01BM Subsection 6.5. Differentiability
• 01BN Proposition Proposition 6.12 .
• 01BP Proof Proof.
• 01BQ Section 7. Envelopes and differentiability
• 01BR Definition Definition 7.1 .
• 01BS Theorem Theorem 7.2 .
• 01BT Corollary Corollary 7.3 .
• 01BU Proof Proof.
• 01BV Proof Proof of Theorem 7.2 .
• 01BW Remark Remark 7.4 .
• 01BX Section 8. The Monge-Ampère equation
• 01BY Theorem Theorem 8.1 .
• 01BZ Subsection 8.1. Uniqueness
• 01C0 Proposition Proposition 8.2 .
• 01C1 Proof Proof.
• 01C2 Subsection 8.2. Existence
• 01C3 Subsection 8.3. Continuity
• 01C4 Lemma Lemma 8.3 .
• 01C5 Proof Proof.
• 01C6 Lemma Lemma 8.4 .
• 01C7 Proof Proof.
• 01C8 Subsection 8.4. An alternative approach
• 01C9 Lemma Lemma 8.5 .
• 01CA Proof Proof.
• 01CB Proposition Proposition 8.6 .
• 01CC Proof Proof.
• 01CD Remark Remark 8.7 .
• 01CE Remark Remark 8.8 .
• 01CF Section 9. Curves and toric varieties
• 01CG Subsection 9.1. Curves
• 01CH Subsection 9.2. Toric varieties
• 01CI Proposition Proposition 9.1 .
• 01CJ Proposition Proposition 9.2 .
• 01CK Proof Proof.
• 01CL Remark Remark 9.3 .
• 01CM Section Appendix A Orthogonality
• 01CN Lemma Lemma A.1 .
• 01CP Proof Proof.
• 01CQ Lemma Lemma A.2 .
• 01CR Proof Proof.
• 01CS Lemma Lemma A.3 .
• 01CT Proof Proof.
• 01CU Theorem Theorem A.4 .
• 01CV Proof Proof of Theorem A.4 .
• 01CW Lemma Lemma A.5 .
• 01CX Proof Proof.
• 01CY Lemma Lemma A.6 .
• 01CZ Proof Proof.
• 01D0 Section Introduction
• 01D1 Section 1. Metrics on lines bundles
• 01D2 Section 2. The Monge-Ampère operator
• 01D3 Section 3. The complex Monge-Ampère equation
• 01D4 Theorem Theorem 3.1 .
• 01D5 Section 4. The non-Archimedean Monge-Ampère equation
• 01D6 Theorem Theorem 4.1 .
• 01D7 Section 5. A variational approach
• 01D8 Section 6. Singular semipositive metrics
• 01D9 Theorem Theorem 6.1 .
• 01DA Theorem Theorem 6.2 .
• 01DB Section 7. Energy
• 01DC Section 8. Envelopes, differentiability and orthogonality
• 01DD Theorem Theorem 8.1 .
• 01DE Theorem Theorem 8.2 .
• 01DF Section 9. Curves
• 01DG Section 10. Toric varieties
• 01DH Section 11. Outlook
• 01DI Section Introduction
• 01DJ Definition Definition .
• 01DK Theorem Theorem A .
• 01DL Theorem Theorem B .
• 01DM Theorem Theorem C .
• 01DN Acknowledgement Acknowledgment .
• 01DP Section 1. Models of varieties over discrete valuation fields
• 01DQ Subsection 1.1. S -varieties
• 01DR Definition Definition 1.1 .
• 01DS Subsection 1.2. Numerical classes and positivity
• 01DT Lemma Lemma 1.2 .
• 01DU Proof Proof.
• 01DV Definition Definition 1.3 .
• 01DW Lemma Lemma 1.4 .
• 01DX Proof Proof.
• 01DY Corollary Corollary 1.5 .
• 01DZ Lemma Lemma 1.6 .
• 01E0 Proof Proof.
• 01E1 Section 2. Projective Berkovich spaces and model functions
• 01E2 Subsection 2.1. Analytifications
• 01E3 Subsection 2.2. Models
• 01E4 Subsection 2.3. Model functions
• 01E5 Definition Definition 2.1 .
• 01E6 Proposition Proposition 2.2 .
• 01E7 Proof Proof.
• 01E8 Corollary Corollary 2.3 .
• 01E9 Corollary Corollary 2.4 .
• 01EA Proof Proof.
• 01EB Corollary Corollary 2.5 .
• 01EC Proof Proof.
• 01ED Section 3. Dual complexes
• 01EE Subsection 3.1. The dual complex of an SNC model
• 01EF Subsection 3.2. Embedding the dual complex in the Berkovich space
• 01EG Theorem Theorem 3.1 .
• 01EH Corollary Corollary 3.2 .
• 01EI Proof Proof.
• 01EJ Definition Definition 3.3 .
• 01EK Lemma Lemma 3.4 .
• 01EL Proof Proof.
• 01EM Proposition Proposition 3.5 .
• 01EN Proof Proof.
• 01EP Definition Definition 3.6 .
• 01EQ Corollary Corollary 3.7 .
• 01ER Proof Proof.
• 01ES Subsection 3.3. Proof of Theorem 3.1
• 01ET Remark Remark 3.8 .
• 01EU Subsection 3.4. Functions on dual complexes
• 01EV Proposition Proposition 3.9 .
• 01EW Proof Proof.
• 01EX Definition Definition 3.10 .
• 01EY Subsection 3.5. Subdivisions and vertical blowups
• 01EZ Theorem Theorem 3.11 .
• 01F0 Proof Proof.
• 01F1 Lemma Lemma 3.12 .
• 01F2 Proof Proof.
• 01F3 Corollary Corollary 3.13 .
• 01F4 Proof Proof.
• 01F5 Section 4. Metrics on line bundles and closed ( 1 , 1 ) -forms
• 01F6 Subsection 4.1. Metrics
• 01F7 Subsection 4.2. Closed ( 1 , 1 ) -forms
• 01F8 Definition Definition 4.1 .
• 01F9 Remark Remark 4.2 .
• 01FA Theorem Theorem 4.3 .
• 01FB Remark Remark 4.4 .
• 01FC Corollary Corollary 4.5 .
• 01FD Proof Proof of Theorem 4.3 .
• 01FE Section 5. Positivity of forms and metrics
• 01FF Subsection 5.1. Positive closed ( 1 , 1 ) -forms and metrics
• 01FG Definition Definition 5.1 .
• 01FH Proposition Proposition 5.2 .
• 01FI Proof Proof.
• 01FJ Corollary Corollary 5.3 .
• 01FK Remark Remark 5.4 .
• 01FL Subsection 5.2. θ -psh model functions
• 01FM Definition Definition 5.5 .
• 01FN Lemma Lemma 5.6 .
• 01FP Proof Proof.
• 01FQ Lemma Lemma 5.7 .
• 01FR Proof Proof.
• 01FS Proposition Proposition 5.8 .
• 01FT Proof Proof.
• 01FU Proposition Proposition 5.9 .
• 01FV Proof Proof.
• 01FW Proposition Proposition 5.10 .
• 01FX Proof Proof.
• 01FY Subsection 5.3. Closedness of θ -psh model functions
• 01FZ Theorem Theorem 5.11 .
• 01G0 Lemma Lemma 5.12 .
• 01G1 Proof Proof.
• 01G2 Proof Proof of Theorem 5.11 .
• 01G3 Remark Remark 5.13 .
• 01G4 Subsection 5.4. Comparison of terminology
• 01G5 Section 6. Equicontinuity
• 01G6 Theorem Theorem 6.1 .
• 01G7 Corollary Corollary 6.2 .
• 01G8 Subsection 6.1. Bounding the values on vertices
• 01G9 Subsection 6.2. Special subdivisions
• 01GA Subsection 6.3. Bounding Lipschitz constants
• 01GB Proposition Proposition 6.3 .
• 01GC Proof Proof of Proposition 6.3 .
• 01GD Lemma Lemma 6.4 .
• 01GE Lemma Lemma 6.5 .
• 01GF Proof Proof of Lemma 6.4 .
• 01GG Proof Proof of Lemma 6.5 .
• 01GH Lemma Lemma 6.6 .
• 01GI Proof Proof.
• 01GJ Section 7. General θ -psh functions and semipositive singular metrics
• 01GK Definition Definition 7.1 .
• 01GL Remark Remark 7.2 .
• 01GM Definition Definition 7.3 .
• 01GN Subsection 7.1. Basic properties
• 01GP Proposition Proposition 7.4 .
• 01GQ Proposition Proposition 7.5 .
• 01GR Proposition Proposition 7.6 .
• 01GS Proof Proof.
• 01GT Subsection 7.2. Equicontinuity
• 01GU Corollary Corollary 7.7 .
• 01GV Subsection 7.3. Compactness
• 01GW Theorem Theorem 7.8 .
• 01GX Proof Proof.
• 01GY Subsection 7.4. Upper envelopes
• 01GZ Theorem Theorem 7.9 .
• 01H0 Lemma Lemma 7.10 .
• 01H1 Proof Proof.
• 01H2 Proof Proof of Theorem 7.9 .
• 01H3 Section 8. Envelopes and regularization
• 01H4 Subsection 8.1. Regularity of envelopes
• 01H5 Definition Definition 8.1 .
• 01H6 Proposition Proposition 8.2 .
• 01H7 Proof Proof.
• 01H8 Theorem Theorem 8.3 .
• 01H9 Lemma Lemma 8.4 .
• 01HA Proof Proof.
• 01HB Proof Proof of Theorem 8.3 .
• 01HC Theorem Theorem 8.5 .
• 01HD Proof Proof of Theorem 8.5 .
• 01HE Corollary Corollary 8.6 .
• 01HF Proof Proof.
• 01HG Subsection 8.2. Monotone regularization of θ -psh functions
• 01HH Theorem Theorem 8.7 .
• 01HI Corollary Corollary 8.8 .
• 01HJ Proof Proof.
• 01HK Lemma Lemma 8.9 .
• 01HL Proof Proof.
• 01HM Section Appendix A Lipschitz constants of convex functions
• 01HN Proposition Proposition A.1 .
• 01HP Proof Proof.
• 01HQ Lemma Lemma A.2 .
• 01HR Proof Proof.
• 01HS Section Appendix B Multiplier ideals on S -varieties
• 01HT Subsection B.1. Kodaira vanishing
• 01HU Theorem Theorem B.1 (Kodaira vanishing) .
• 01HV Proof Proof.
• 01HW Subsection B.2. Kawamata-Viehweg vanishing
• 01HX Lemma Lemma B.2 (Covering trick) .
• 01HY Proof Proof.
• 01HZ Theorem Theorem B.3 (Kawamata-Viehweg vanishing) .
• 01I0 Proof Proof.
• 01I1 Subsection B.3. Multiplier ideals
• 01I2 Definition Definition B.4 .
• 01I3 Theorem Theorem B.5 (Nadel Vanishing) .
• 01I4 Proof Proof.
• 01I5 Lemma Lemma B.6 (Local vanishing) .
• 01I6 Proof Proof.
• 01I7 Theorem Theorem B.7 (Subadditivity) .
• 01I8 Proof Proof.
• 01I9 Theorem Theorem B.8 (Uniform generation property) .
• 01IA Proof Proof.
• 01IB Paragraph Acknowledgments
• 01IC Section 1. Metrized line bundles and measures
• 01ID Subsection 1.1. Continuous metrics
• 01IE Paragraph Definition
• 01IF Paragraph The Abelian group of metrized line bundles
• 01IG Subsection 1.2. The case of complex analytic spaces
• 01IH Paragraph Smooth metrics
• 01II Paragraph Curvature
• 01IJ Paragraph Products, measures
• 01IK Paragraph The Poincaré–Lelong equation
• 01IL Paragraph Archimedean height pairing
• 01IM Paragraph Positivity
• 01IN Paragraph Semi-positive continuous metrics
• 01IP Paragraph Admissible metrics
• 01IQ Paragraph Local height pairing (admissible case)
• 01IR Subsection 1.3. The case of non-archimedean analytic spaces
• 01IS Paragraph Continuous metrics
• 01IT Paragraph Smooth metrics
• 01IU Paragraph Green functions ; smooth functions
• 01IV Paragraph Example : projective space
• 01IW Paragraph The Abelian group of smooth line bundles
• 01IX Paragraph Semi-positive metrics
• 01IY Paragraph Continuous semi-positive metrics
• 01IZ Paragraph Measures (smooth metrics)
• 01J0 Paragraph Local height pairing (admissible metrics)
• 01J1 Paragraph Measures (admissible metrics)
• 01J2 Paragraph Integrating Green functions
• 01J3 Proposition Proposition 1.3.2 .
• 01J4 Proof Démonstration.
• 01J5 Section 2. Examples
• 01J6 Subsection 2.1. The projective space
• 01J7 Proposition Proposition 2.1.1 .
• 01J8 Subsection 2.2. Semi-stable curves and reduction graphs
• 01J9 Paragraph The reduction graph of the special fiber
• 01JA Paragraph Drawing the reduction graph on the Berkovich space
• 01JB Paragraph Metrized line bundles and the reduction graph
• 01JC Lemma Lemma 2.2.2 .
• 01JD Proof Démonstration.
• 01JE Subsection 2.3. Local character of the measures
• 01JF Definition Definition 2.3.1 .
• 01JG Definition Definition 2.3.2 .
• 01JH Proposition Proposition 2.3.3 .
• 01JI Proof Démonstration.
• 01JJ Subsection 2.4. Polarized dynamical systems
• 01JK Lemma Lemma 2.4.1 .
• 01JL Proof Démonstration.
• 01JM Paragraph The canonical measure
• 01JN Paragraph The Fatou set
• 01JP Proposition Proposition 2.4.2 .
• 01JQ Proof Démonstration.
• 01JR Corollary Corollary 2.4.3 .
• 01JS Proof Démonstration.
• 01JT Paragraph Remarks
• 01JU Subsection 2.5. Abelian varieties
• 01JV Paragraph Canonical metrics
• 01JW Paragraph The case of good reduction
• 01JX Paragraph Gubler’s description
• 01JY Theorem Theorem 2.5.1 .
• 01JZ Section 3. Applications to Arakelov geometry
• 01K0 Subsection 3.1. Adelic metrics and heights
• 01K1 Paragraph Adelic metrics
• 01K2 Paragraph Heights
• 01K3 Paragraph Heights of points
• 01K4 Paragraph Zhang’s inequality
• 01K5 Subsection 3.2. Mahler measures and heights of divisors
• 01K6 Paragraph Paragraph
• 01K7 Paragraph Néron–Tate heights
• 01K8 Paragraph Canonical measures
• 01K9 Paragraph Superelliptic curves
• 01KA Subsection 3.3. An equidistribution theorem
• 01KB Paragraph Bogomolov’s conjecture
• 01KC Paragraph Paragraph
• 01KD Theorem Theorem 3.3.1 .
• 01KE Subsection 3.4. Lower bounds for heights and the Hodge index theorem
• 01KF Paragraph The arithmetic Hodge index theorem
• 01KG Proposition Proposition 3.4.1 .
• 01KH Proof Démonstration.
• 01KI Paragraph Paragraph
• 01KJ Paragraph An example
• 01KK Paragraph Application to dynamical systems
• 01KL Proposition Proposition 3.4.3 .
• 01KM Proof Démonstration.
• 01KN Paragraph Remarks
• 01KP Section Introduction
• 01KQ Subsection 0.1 Contexte et philosophie
• 01KR Subsubsection 0.1.1
• 01KS Subsubsection 0.1.2
• 01KT Subsection 0.2 Cet article
• 01KU Subsubsection 0.2.1
• 01KV Subsubsection 0.2.2 Fonctions et formes lisses
• 01KW Subsubsection 0.2.3 Courants
• 01KX Subsubsection 0.2.4
• 01KY Subsubsection 0.2.5
• 01KZ Subsection 0.3 Géométrie tropicale
• 01L0 Subsubsection 0.3.1
• 01L1 Subsubsection 0.3.2
• 01L2 Subsubsection 0.3.3
• 01L3 Subsubsection 0.3.4
• 01L4 Subsubsection 0.3.5
• 01L5 Subsection 0.4 Remarques, perspectives
• 01L6 Subsubsection 0.4.1
• 01L7 Subsubsection 0.4.2
• 01L8 Subsubsection 0.4.3
• 01L9 Subsection 0.5 Remerciements
• 01LA Section § 1 Formes différentielles en géométrie tropicale
• 01LB Subsection 1.1 Polytopes
• 01LC Subsubsection 1.1.1 Cellules
• 01LD Subsubsection 1.1.2 Polytopes
• 01LE SourceObject \exemname 1.1.3 .
• 01LF Subsubsection 1.1.4
• 01LG Subsubsection 1.1.5
• 01LH Subsection 1.2 Superformes, d’après A. Lagerberg
• 01LI Subsubsection 1.2.1
• 01LJ Subsubsection 1.2.2
• 01LK Subsubsection 1.2.3
• 01LL Subsubsection 1.2.4
• 01LM Subsubsection 1.2.5
• 01LN Subsubsection 1.2.6
• 01LP Subsubsection 1.2.7
• 01LQ Subsubsection 1.2.8
• 01LR Subsubsection 1.2.9
• 01LS SourceObject \lemmname 1.2.10 .
• 01LT Proof Démonstration.
• 01LU Subsubsection 1.2.11
• 01LV SourceObject \remaname 1.2.12 .
• 01LW Subsection 1.3 Intégrale d’une ( p , n ) -forme sur un sous-espace de dimension p
• 01LX Subsubsection 1.3.1
• 01LY Subsubsection 1.3.2 Non-canonicité des intégrales de Lagerberg
• 01LZ Subsubsection 1.3.3
• 01M0 Subsubsection 1.3.4
• 01M1 Subsubsection 1.3.5 Le cas des formes de type ( n , n ) : les vecteurs-volume
• 01M2 Subsubsection 1.3.6
• 01M3 Subsubsection 1.3.7 Le cas des ( n − 1 , n ) -formes : les intégrales de bord
• 01M4 SourceObject \lemmname 1.3.8 (Formule de Green) .
• 01M5 Proof Démonstration.
• 01M6 Subsection 1.4 Formes différentielles sur un polytope
• 01M7 Subsubsection 1.4.1
• 01M8 SourceObject \remaname 1.4.2 .
• 01M9 Subsubsection 1.4.3
• 01MA Subsubsection 1.4.4
• 01MB SourceObject \lemmname 1.4.5 .
• 01MC Proof Démonstration.
• 01MD Subsubsection 1.4.6
• 01ME Subsubsection 1.4.7
• 01MF Subsection 1.5 Calibrages et intégrales sur les polytopes
• 01MG Subsubsection 1.5.1 Intégrale d’une ( n , n ) -forme intégrable sur un polytope calibré
• 01MH Subsubsection 1.5.2
• 01MI Subsubsection 1.5.3 Discordance
• 01MJ Subsubsection 1.5.4 Intégrale de bord
• 01MK Subsubsection 1.5.5
• 01ML Subsubsection 1.5.6
• 01MM SourceObject \lemmname 1.5.7 (Formules de Stokes et Green) .
• 01MN Proof Démonstration.
• 01MP Subsubsection 1.5.8 Fonctorialité
• 01MQ Section § 2 Géométrie analytique et géométrie tropicale
• 01MR Subsection 2.1 Géométrie analytique : conventions et notations
• 01MS Subsubsection 2.1.1
• 01MT Subsubsection 2.1.2
• 01MU Subsubsection 2.1.3 Dimension centrale d’un germe d’espace k -analytique
• 01MV Subsubsection 2.1.4
• 01MW Subsubsection 2.1.5
• 01MX SourceObject \lemmname 2.1.6 .
• 01MY Proof Démonstration.
• 01MZ Subsection 2.2 Tores, squelettes, tropicalisations, moments
• 01N0 Subsubsection 2.2.1
• 01N1 Subsubsection 2.2.2
• 01N2 Subsubsection 2.2.3 Tropicalisations
• 01N3 Subsubsection 2.2.4 Squelette de 𝐆 m n
• 01N4 Subsubsection 2.2.5 Squelette d’un tore
• 01N5 SourceObject \definame 2.2.6 .
• 01N6 SourceObject \remaname 2.2.7 .
• 01N7 Subsection 2.3 Les tropicalisations sont des polytopes
• 01N8 Subsubsection 2.3.1 Une convention.
• 01N9 Subsubsection 2.3.2 Tropicalisations : le cas global.
• 01NA Subsubsection 2.3.3 Tropicalisations : le cas local
• 01NB Subsubsection 2.3.4 Images réciproques du squelette
• 01NC Subsection 2.4 Squelettes, degrés et tropicalisations
• 01ND Subsubsection 2.4.1 Squelettes et isogénies
• 01NE Subsubsection 2.4.2 Morphisme finis et plats au-dessus d’un sous-ensemble du but
• 01NF Subsubsection 2.4.3 Un exemple
• 01NG Subsubsection 2.4.4 Degrés et tropicalisations
• 01NH Section § 3 Formes différentielles réelles en géométrie ultramétrique
• 01NI Subsection 3.1 Formes de type ( p , q ) sur un espace analytique
• 01NJ Subsubsection 3.1.1
• 01NK Subsubsection 3.1.2
• 01NL Subsubsection 3.1.3
• 01NM Subsubsection 3.1.4
• 01NN Subsubsection 3.1.5
• 01NP Subsubsection 3.1.6
• 01NQ Subsubsection 3.1.7
• 01NR SourceObject \lemmname 3.1.8 .
• 01NS Proof Démonstration.
• 01NT Subsubsection 3.1.9
• 01NU SourceObject \lemmname 3.1.10 .
• 01NV Proof Démonstration.
• 01NW Subsection 3.2 Support
• 01NX Subsubsection 3.2.1
• 01NY SourceObject \lemmname 3.2.2 .
• 01NZ Proof Démonstration.
• 01P0 SourceObject \coroname 3.2.3 .
• 01P1 Proof Démonstration.
• 01P2 SourceObject \coroname 3.2.4 .
• 01P3 Proof Démonstration.
• 01P4 SourceObject \lemmname 3.2.5 .
• 01P5 Proof Démonstration.
• 01P6 SourceObject \lemmname 3.2.6 .
• 01P7 Proof Démonstration.
• 01P8 Proposition \propname 3.2.7 .
• 01P9 Proof Démonstration.
• 01PA Proposition \propname 3.2.8 .
• 01PB Proof Démonstration.
• 01PC SourceObject \lemmname 3.2.9 .
• 01PD Proof Démonstration.
• 01PE Subsection 3.3 Partitions de l’unité
• 01PF SourceObject \lemmname 3.3.1 .
• 01PG Proof Démonstration.
• 01PH SourceObject \coroname 3.3.2 .
• 01PI Proof Démonstration.
• 01PJ SourceObject \coroname 3.3.3 .
• 01PK Proof Démonstration.
• 01PL SourceObject \coroname 3.3.4 .
• 01PM Proof Démonstration.
• 01PN Proposition \propname 3.3.5 (Stone–Weierstraß) .
• 01PP Proof Démonstration.
• 01PQ Proposition \propname 3.3.6 (Partitions de l’unité lisses) .
• 01PR Proof Démonstration.
• 01PS SourceObject \coroname 3.3.7 .
• 01PT Subsection 3.4 Existence de tropicalisations (presque) globales
• 01PU Proposition \propname 3.4.1 .
• 01PV Proof Démonstration.
• 01PW Subsubsection 3.4.2 Bref rappel sur les réductions graduées à la Temkin.
• 01PX SourceObject \lemmname 3.4.3 .
• 01PY Proof Démonstration.
• 01PZ Proposition \propname 3.4.4 .
• 01Q0 Proof Démonstration.
• 01Q1 Subsection 3.5 Calibrage canonique d’une tropicalisation
• 01Q2 Subsubsection 3.5.1
• 01Q3 SourceObject \lemmname 3.5.2 .
• 01Q4 Proof Démonstration.
• 01Q5 Subsection 3.6 Condition d’harmonie
• 01Q6 SourceObject \theoname 3.6.1 .
• 01Q7 Proof Démonstration.
• 01Q8 Subsection 3.7 Intégrales et intégrales de bord des formes tropicales
• 01Q9 Subsubsection 3.7.1
• 01QA Proposition \propname 3.7.2 .
• 01QB Proposition \propname 3.7.3 .
• 01QC Proof Démonstration.
• 01QD SourceObject \lemmname 3.7.4 .
• 01QE Proof Démonstration.
• 01QF SourceObject \remaname 3.7.5 .
• 01QG Subsection 3.8 Intégrales et intégrales de bord sur les domaines d’un bon espace k -analytique
• 01QH Subsubsection 3.8.1
• 01QI Subsubsection 3.8.2
• 01QJ Subsubsection 3.8.3
• 01QK SourceObject \lemmname 3.8.4 .
• 01QL Proof Démonstration.
• 01QM Subsubsection 3.8.5 Compatibilité avec les définitions précédentes
• 01QN Subsection 3.9 Premières propriétés de l’intégrale
• 01QP Subsubsection 3.9.1
• 01QQ Subsubsection 3.9.2
• 01QR SourceObject \remaname 3.9.3 .
• 01QS Subsubsection 3.9.4
• 01QT Subsection 3.10 Propriétés spécifiques aux intégrales de ( n , n ) -formes
• 01QU Subsubsection 3.10.1
• 01QV Subsubsection 3.10.2
• 01QW Subsubsection 3.10.3
• 01QX Subsubsection 3.10.4
• 01QY Proposition \propname 3.10.5 .
• 01QZ Proof Démonstration.
• 01R0 SourceObject \coroname 3.10.6 .
• 01R1 Proposition \propname 3.10.7 .
• 01R2 Proof Démonstration.
• 01R3 Subsection 3.11 Positivité, régularité et continuité
• 01R4 Proposition \propname 3.11.1 .
• 01R5 Proof Démonstration.
• 01R6 Proposition \propname 3.11.2 .
• 01R7 Proof Démonstration.
• 01R8 Subsection 3.12 Formules de Stokes et de Green
• 01R9 SourceObject \theoname 3.12.1 .
• 01RA SourceObject \theoname 3.12.2 .
• 01RB Proof Démonstration.
• 01RC Section § 4 Courants
• 01RD Subsection 4.1 Topologie sur l’espace des formes
• 01RE Subsubsection 4.1.1
• 01RF SourceObject \lemmname 4.1.2 .
• 01RG Proof Démonstration.
• 01RH Subsection 4.2 Courants
• 01RI Subsubsection 4.2.1 Formes à support propre
• 01RJ Subsubsection 4.2.2
• 01RK Subsubsection 4.2.3 Image directe d’un courant
• 01RL Subsubsection 4.2.4 Propriété de faisceau
• 01RM SourceObject \lemmname 4.2.5 .
• 01RN Proof Démonstration.
• 01RP Subsection 4.3 Courants d’intégration
• 01RQ Subsubsection 4.3.1
• 01RR Subsubsection 4.3.2
• 01RS Subsubsection 4.3.3
• 01RT Subsubsection 4.3.4
• 01RU Subsubsection 4.3.5
• 01RV SourceObject \lemmname 4.3.6 .
• 01RW Proof Démonstration.
• 01RX SourceObject \coroname 4.3.7 .
• 01RY Proof Démonstration.
• 01RZ SourceObject \remaname 4.3.8 .
• 01S0 SourceObject \remaname 4.3.9 .
• 01S1 Subsection 4.4 Calcul différentiel sur les courants
• 01S2 Subsubsection 4.4.1
• 01S3 Subsubsection 4.4.2
• 01S4 Subsubsection 4.4.3
• 01S5 Subsubsection 4.4.4
• 01S6 Proposition \propname 4.4.5 .
• 01S7 Proof Démonstration.
• 01S8 Subsection 4.5 Multiplicités et degré intégral
• 01S9 Subsubsection 4.5.1
• 01SA SourceObject \lemmname 4.5.2 .
• 01SB Proof Démonstration.
• 01SC Subsubsection 4.5.3
• 01SD Subsubsection 4.5.4
• 01SE Subsubsection 4.5.5
• 01SF Subsubsection 4.5.6
• 01SG Subsubsection 4.5.7
• 01SH SourceObject \exemname 4.5.8 .
• 01SI SourceObject \exemname 4.5.9 .
• 01SJ SourceObject \exemname 4.5.10 .
• 01SK Subsubsection 4.5.11
• 01SL Subsection 4.6 Le courant d’intégration δ div ⁡ ( f ) et la formule de Poincaré–Lelong
• 01SM SourceObject \lemmname 4.6.1 .
• 01SN Proof Démonstration.
• 01SP Subsubsection 4.6.2
• 01SQ Subsubsection 4.6.3
• 01SR SourceObject \lemmname 4.6.4 .
• 01SS Proof Démonstration.
• 01ST SourceObject \theoname 4.6.5 (Formule de Poincaré-Lelong) .
• 01SU Proof Démonstration.
• 01SV Proposition \propname 4.6.6 .
• 01SW Proof Démonstration.
• 01SX Section § 5 Courants positifs, théorie de Bedford-Taylor
• 01SY Subsection 5.1 Formes positives en géométrie tropicale
• 01SZ SourceObject \definame 5.1.1 .
• 01T0 Subsubsection 5.1.2
• 01T1 Proposition \propname 5.1.3 ( [ 43 ] , Proposition 2.2) .
• 01T2 Proposition \propname 5.1.4 ( [ 43 ] , Lemma 2.1) .
• 01T3 Proof Démonstration.
• 01T4 SourceObject \remaname 5.1.5 .
• 01T5 Subsubsection 5.1.6
• 01T6 Subsection 5.2 Formes positives sur un sous-espace linéaire par morceaux
• 01T7 SourceObject \definame 5.2.1 .
• 01T8 Subsubsection 5.2.2
• 01T9 SourceObject \lemmname 5.2.3 .
• 01TA Proof Démonstration.
• 01TB Subsection 5.3 Formes lisses positives sur un espace analytique
• 01TC SourceObject \definame 5.3.1 .
• 01TD Subsubsection 5.3.2
• 01TE SourceObject \lemmname 5.3.3 .
• 01TF Proof Démonstration.
• 01TG Subsection 5.4 Courants positifs
• 01TH Subsubsection 5.4.1
• 01TI Subsubsection 5.4.2
• 01TJ Subsubsection 5.4.3
• 01TK SourceObject \lemmname 5.4.4 .
• 01TL Proof Démonstration.
• 01TM Subsubsection 5.4.5
• 01TN Proposition \propname 5.4.6 .
• 01TP Proof Démonstration.
• 01TQ SourceObject \coroname 5.4.7 .
• 01TR Proof Démonstration.
• 01TS Subsubsection 5.4.8
• 01TT Proposition \propname 5.4.9 .
• 01TU Proof Démonstration.
• 01TV Subsection 5.5 Fonctions plurisousharmoniques
• 01TW SourceObject \definame 5.5.1 .
• 01TX SourceObject \lemmname 5.5.2 .
• 01TY Proof Démonstration.
• 01TZ SourceObject \lemmname 5.5.3 .
• 01U0 Proof Démonstration.
• 01U1 SourceObject \remaname 5.5.4 .
• 01U2 Subsection 5.6 Produits de courants positifs
• 01U3 SourceObject \lemmname 5.6.1 .
• 01U4 Proof Démonstration.
• 01U5 Proposition \propname 5.6.2 .
• 01U6 Proof Démonstration.
• 01U7 SourceObject \definame 5.6.3 .
• 01U8 SourceObject \remaname 5.6.4 .
• 01U9 SourceObject \coroname 5.6.5 .
• 01UA Proof Démonstration.
• 01UB SourceObject \coroname 5.6.6 .
• 01UC SourceObject \definame 5.6.7 .
• 01UD Subsubsection 5.6.8
• 01UE Subsection 5.7 Un calcul d’opérateur de Monge-Ampère (cas réel)
• 01UF SourceObject \lemmname 5.7.1 .
• 01UG Proof Démonstration.
• 01UH SourceObject \lemmname 5.7.2 .
• 01UI Proof Démonstration.
• 01UJ Subsubsection 5.7.3
• 01UK Proposition \propname 5.7.4 .
• 01UL Proof Démonstration.
• 01UM SourceObject \lemmname 5.7.5 .
• 01UN Proof Démonstration.
• 01UP SourceObject \lemmname 5.7.6 .
• 01UQ Proof Démonstration.
• 01UR SourceObject \remaname 5.7.7 .
• 01US Section § 6 Fibrés vectoriels métrisés
• 01UT Subsection 6.1 Compléments sur les espaces de Zariski-Riemann
• 01UU Subsubsection 6.1.1 Espaces de Zariski-Riemann
• 01UV SourceObject \remaname 6.1.2 .
• 01UW Subsubsection 6.1.3
• 01UX SourceObject \definame 6.1.4 .
• 01UY SourceObject \remaname 6.1.5 .
• 01UZ Subsubsection 6.1.6
• 01V0 Subsubsection 6.1.7
• 01V1 Subsubsection 6.1.8 Réduction à la Temkin
• 01V2 Subsubsection 6.1.9
• 01V3 Subsubsection 6.1.10
• 01V4 Subsection 6.2 Métriques
• 01V5 Subsubsection 6.2.1 Fibrés vectoriels
• 01V6 SourceObject \definame 6.2.2 (Métriques) .
• 01V7 SourceObject \remaname 6.2.3 .
• 01V8 SourceObject \definame 6.2.4 (Métriques lisses) .
• 01V9 SourceObject \remaname 6.2.5 .
• 01VA Proposition \propname 6.2.6 .
• 01VB Proof Démonstration.
• 01VC Subsubsection 6.2.7
• 01VD SourceObject \lemmname 6.2.8 .
• 01VE Proof Démonstration.
• 01VF SourceObject \definame 6.2.9 .
• 01VG SourceObject \exemname 6.2.10 (Métriques formelles) .
• 01VH Subsubsection 6.2.11
• 01VI SourceObject \remaname 6.2.12 .
• 01VJ Proposition \propname 6.2.13 .
• 01VK Proof Démonstration.
• 01VL Subsubsection 6.2.14
• 01VM Subsubsection 6.2.15
• 01VN Subsection 6.3 Métriques approchables
• 01VP Subsubsection 6.3.1
• 01VQ Proposition \propname 6.3.2 .
• 01VR Proof Démonstration.
• 01VS SourceObject \coroname 6.3.3 .
• 01VT Proof Démonstration.
• 01VU SourceObject \coroname 6.3.4 .
• 01VV Proof Démonstration.
• 01VW SourceObject \coroname 6.3.5 .
• 01VX Proof Démonstration.
• 01VY Subsection 6.4 Courant de courbure d’un fibré métrisé
• 01VZ Subsubsection 6.4.1
• 01W0 Subsubsection 6.4.2
• 01W1 Proposition \propname 6.4.3 .
• 01W2 Proof Démonstration.
• 01W3 SourceObject \coroname 6.4.4 .
• 01W4 Proof Démonstration.
• 01W5 SourceObject \remaname 6.4.5 .
• 01W6 Subsection 6.5 Préliminaires à propos de la réduction des germes
• 01W7 SourceObject \lemmname 6.5.1 .
• 01W8 Proof Démonstration.
• 01W9 SourceObject \remaname 6.5.2 .
• 01WA Subsubsection 6.5.3
• 01WB Subsubsection 6.5.4 Réduction de cocycles
• 01WC Subsubsection 6.5.5 Réduction et relèvement de cocycle
• 01WD Subsection 6.6 Fibré résiduel d’un fibré PL
• 01WE Subsubsection 6.6.1
• 01WF Subsubsection 6.6.2 Liens entre un fibré métrisé PL et son fibré résiduel
• 01WG Subsubsection 6.6.3 Sections du fibré résiduel
• 01WH SourceObject \lemmname 6.6.4 .
• 01WI Proof Démonstration.
• 01WJ Subsubsection 6.6.5 Fibré métrisé PL de fibré résiduel prescrit
• 01WK Subsubsection 6.6.6
• 01WL Subsubsection 6.6.7 Fonctorialité des constructions
• 01WM SourceObject \exemname 6.6.8 (Le cas d’un fibré formel) .
• 01WN SourceObject \exemname 6.6.9 (Fibré sur une variété propre en valuation triviale) .
• 01WP Subsection 6.7 Variation de la dimension de définition simultanée d’une famille de fibrés métrisés PL
• 01WQ Subsubsection 6.7.1
• 01WR SourceObject \definame 6.7.2 .
• 01WS Proposition \propname 6.7.3 .
• 01WT Proof Démonstration.
• 01WU Subsection 6.8 Positivité résiduelle et « concavité » de la norme d’une section
• 01WV Subsubsection 6.8.1
• 01WW Proposition \propname 6.8.2 .
• 01WX Proof Démonstration.
• 01WY Proposition \propname 6.8.3 .
• 01WZ Proof Démonstration.
• 01X0 Subsection 6.9 Calcul de l’opérateur de Monge-Ampère (métriques formelles)
• 01X1 Subsubsection 6.9.1
• 01X2 Proposition \propname 6.9.2 .
• 01X3 Proof Démonstration.
• 01X4 SourceObject \theoname 6.9.3 .
• 01X5 Proof Démonstration.
• 01X6 Proposition \propname 6.9.4 .
• 01X7 Proof Démonstration.
• 01X8 SourceObject \lemmname 6.9.5 .
• 01X9 Proof Démonstration.
• 01XA Subsubsection 6.9.6
• 01XB Proposition \propname 6.9.7 .
• 01XC Proof Démonstration.
• 01XD Section 1. Introduction
• 01XE Theorem Theorem 1.1 .
• 01XF Definition Definition 1.2 .
• 01XG Theorem Theorem 1.3 .
• 01XH Remark Remark 1.1 .
• 01XI Theorem Theorem 1.4 .
• 01XJ Theorem Theorem 1.5 .
• 01XK Remark Remark 1.2 .
• 01XL Subsection 1.1. Outline of the proof Theorem 1.1 , the codimension 4 conjecture
• 01XM Definition Definition 1.6 .
• 01XN Lemma Lemma 1.7 ( [ ChCo1 ] ) .
• 01XP Theorem Theorem 1.8 .
• 01XQ Theorem Theorem 1.9 .
• 01XR Definition Definition 1.10 .
• 01XS Theorem Theorem 1.11 .
• 01XT Section 2. Background and Preliminaries
• 01XU Subsection 2.1. Stratification of Limit Spaces
• 01XV Definition Definition 2.1 .
• 01XW Subsection 2.2. ϵ -Regularity Theorems
• 01XX Definition Definition 2.2 .
• 01XY Theorem Theorem 2.3 ( [ A90 ] , [ ChCo1 ] ) .
• 01XZ Subsection 2.3. Examples
• 01Y0 Example Example 2.1 .
• 01Y1 Example Example 2.2 .
• 01Y2 Example Example 2.3 .
• 01Y3 Section 3. Proof of the Transformation Theorem
• 01Y4 Subsection 3.1. Higher Order Estimates
• 01Y5 Proof Proof of Theorem 1.9 .
• 01Y6 Subsection 3.2. Proof of the Transformation theorem
• 01Y7 Proof Proof of Theorem 1.11 :
• 01Y8 Lemma Lemma 3.1 .
• 01Y9 Remark Remark 3.1 .
• 01YA Remark Remark 3.2 .
• 01YB Section 4. Proof of Theorem 1.8 , the Slicing Theorem
• 01YC Lemma Lemma 4.1 .
• 01YD Proof Proof.
• 01YE Lemma Lemma 4.2 .
• 01YF Proof Proof.
• 01YG Section 5. Codimension 4 Regularity of Singular Limits
• 01YH Subsection 5.1. Nonexistence of Codimension 2 Singularities
• 01YI Theorem Theorem 5.1 ( ( n − 2 ) -Symmetric Limits) .
• 01YJ Proof Proof of Theorem 5.1 .
• 01YK Corollary Corollary 5.2 .
• 01YL Proof Proof.
• 01YM Subsection 5.2. Nonexistence of Codimension 3 singularities
• 01YN Theorem Theorem 5.3 ( ( n − 3 ) -Symmetric Limits) .
• 01YP Proof Proof.
• 01YQ Subsection 5.3. Proof of Hausdorff Estimates of Theorem 1.1
• 01YR Section 6. The ϵ -regularity Theorem
• 01YS Theorem Theorem 6.1 .
• 01YT Proof Proof.
• 01YU Section 7. Quantitative Stratification and Effective Estimates
• 01YV Definition Definition 7.1 .
• 01YW Definition Definition 7.2 .
• 01YX Theorem Theorem 7.3 (Quantitative Stratification, [ ChNa13 ] ) .
• 01YY Subsection 7.1. Proof of Theorem 1.3
• 01YZ Proof Proof.
• 01Z0 Subsection 7.2. L q -Estimates for Harmonic Functions on Einstein Manifolds
• 01Z1 Theorem Theorem 7.4 .
• 01Z2 Proof Proof.
• 01Z3 Section 8. Improved Estimates in Dimension 4
• 01Z4 Subsection 8.1. Diffeomorphisms and Harmonic Radius
• 01Z5 Theorem Theorem 8.1 .
• 01Z6 Theorem Theorem 8.2 .
• 01Z7 Subsection 8.2. Annulus Estimates
• 01Z8 Theorem Theorem 8.3 .
• 01Z9 Proof Proof.
• 01ZA Subsection 8.3. Regularity Scale Estimates
• 01ZB Definition Definition 8.4 .
• 01ZC Remark Remark 8.1 .
• 01ZD Lemma Lemma 8.5 .
• 01ZE Proof Proof.
• 01ZF Corollary Corollary 8.6 .
• 01ZG Proof Proof.
• 01ZH Proof Proof of Estimate ( 1.9 ) of Theorem 1.5 .
• 01ZI Subsection 8.4. Finite Diffeomorphism Type
• 01ZJ Definition Definition 8.7 .
• 01ZK Lemma Lemma 8.8 .
• 01ZL Proof Proof.
• 01ZM Lemma Lemma 8.9 .
• 01ZN Proof Proof.
• 01ZP Lemma Lemma 8.10 .
• 01ZQ Proof Proof.
• 01ZR Theorem Theorem 8.11 .
• 01ZS Proof Proof.
• 01ZT Proof Proof of Theorem 1.4 .
• 01ZU Subsection 8.5. L 2 Curvature Estimates
• 01ZV Theorem Theorem 8.12 .
• 01ZW Proof Proof.
• 01ZX Proof Proof of Theorem 1.5 .
• 01ZY Section 9. Conjectures
• 01ZZ Conjecture Conjecture 9.1 .
• 0200 Conjecture Conjecture 9.2 .
• 0201 Section 1. Mirror symmetry before SYZ
• 0202 Section 2. Formulation of the SYZ conjecture
• 0203 Conjecture Conjecture 2.1 (The SYZ conjecture [ 153 ] ) .
• 0204 Section 3. Semi-flat SYZ
• 0205 Section 4. Constructing SYZ fibrations
• 0206 Section 5. SYZ for compact Calabi-Yau manifolds
• 0207 Section 6. SYZ for noncompact Calabi-Yau manifolds
• 0208 Section 7. SYZ in the non–Calabi-Yau setting
• 0209 Section 8. Beyond SYZ
• 020A Section 1. Introduction
• 020B Conjecture Conjecture 1.1 .
• 020C Conjecture Conjecture 1.2 (Yau-Tian-Donaldson) .
• 020D Theorem Theorem 1.1 .
• 020E Corollary Corollary 1.1 .
• 020F Proposition Proposition 1.2 .
• 020G Theorem Theorem 1.2 .
• 020H Remark Remark 1.3 .
• 020I Theorem Theorem 1.3 .
• 020J Theorem Theorem 1.4 .
• 020K Proposition Proposition 1.4 .
• 020L Theorem Theorem 1.5 .
• 020M Theorem Theorem 1.6 .
• 020N Theorem Theorem 1.7 .
• 020P Theorem Theorem 1.8 .
• 020Q Corollary Corollary 1.5 .
• 020R Section 2. The volume ratio ω φ n ω n and C 1 bound on Kähler potential
• 020S Proposition Proposition 2.1 .
• 020T Proof Proof.
• 020U Theorem Theorem 2.1 .
• 020V Proof Proof.
• 020W Theorem Theorem 2.2 .
• 020X Proof Proof.
• 020Y Section 3. The volume ratio ω φ n ω 0 n and W 2 , p bound on Kähler potential
• 020Z Theorem Theorem 3.1 .
• 0210 Proof Proof.
• 0211 Remark Remark 3.1 .
• 0212 Corollary Corollary 3.2 .
• 0213 Section 4. C 1 , 1 bound of the Kähler potential in terms of its W 2 , p bound
• 0214 Theorem Theorem 4.1 .
• 0215 Corollary Corollary 4.1 .
• 0216 Proposition Proposition 4.2 .
• 0217 Remark Remark 4.3 .
• 0218 Proof Proof.
• 0219 Section 5. Entropy bound of the volume ratio and C 0 bound of Kähler potential
• 021A Theorem Theorem 5.1 .
• 021B Proposition Proposition 5.1 .
• 021C Theorem Theorem 5.2 .
• 021D Corollary Corollary 5.2 .
• 021E Proof Proof.
• 021F Lemma Lemma 5.3 .
• 021G Corollary Corollary 5.4 .
• 021H Proof Proof.
• 021I Proof Proof.
• 021J Lemma Lemma 5.5 .
• 021K Remark Remark 5.6 .
• 021L Proof Proof.
• 021M Section 6. Some local estimates
• 021N Proposition Proposition 6.1 .
• 021P Corollary Corollary 6.2 .
• 021Q Proof Proof.
• 021R Proof Proof.
• 021S Lemma Lemma 6.3 .
• 021T Proof Proof.
• 021U Lemma Lemma 6.4 .
• 021V Proposition Proposition 6.5 .
• 021W Proof Proof.
• 021X Section 1 Introduction
• 021Y Theorem Theorem 1.1 (Tian-Yau, [ 16 ] ) .
• 021Z Question Question .
• 0220 Theorem Theorem 1.2 .
• 0221 Acknowledgement Acknowledgement .
• 0222 Section 2 Generalized Calabi ansatz and ODE reduction
• 0223 Subsection 2.1 The generalized Calabi ansatz
• 0224 Notation Notation .
• 0225 Subsection 2.2 Semiflat metrics and the Calabi ansatz
• 0226 Subsection 2.3 Simplifications for proportional line bundles
• 0227 Lemma Lemma 2.1 .
• 0228 Remark Remark 2.2 .
• 0229 Subsection 2.4 Relevance to the Tian-Yau problem
• 022A Subsection 2.5 ODE reduction
• 022B Lemma Lemma 2.3 .
• 022C Proof Proof.
• 022D Remark Remark 2.4 .
• 022E Remark Remark 2.5 .
• 022F Remark Remark 2.6 .
• 022G Subsection 2.6 Basic length scales of the generic region
• 022H Subsection 2.7 Boundary condition of the ODE
• 022I Subsubsection Geometric meaning of the boundary condition
• 022J Remark Remark 2.7 .
• 022K Subsection 2.8 Remarks on other related literature
• 022L Subsubsection 2.8.1 Degenerating hypersurfaces
• 022M Subsubsection 2.8.2 Exotic metrics on ℂ n
• 022N Remark Remark 2.8 .
• 022P Remark Remark 2.9 .
• 022Q Remark Remark 2.10 .
• 022R Subsubsection 2.8.3 Non-archimedean meaning?
• 022S Remark Remark 2.11 .
• 022T Section 3 More on the ODE reduction
• 022U Subsection 3.1 Reformulations of the ODE
• 022V Subsubsection First reformulation of the ODE
• 022W Lemma Lemma 3.1 .
• 022X Proof Proof.
• 022Y Remark Remark 3.2 .
• 022Z Remark Remark 3.3 .
• 0230 Subsubsection Second reformulation of the ODE
• 0231 Subsection 3.2 First integral of the ODE
• 0232 Lemma Lemma 3.4 .
• 0233 Proof Proof.
• 0234 Corollary Corollary 3.5 .
• 0235 Proof Proof.
• 0236 Subsection 3.3 Global matching problem
• 0237 Lemma Lemma 3.6 .
• 0238 Proof Proof.
• 0239 Subsection 3.4 Legendre transform
• 023A Lemma Lemma 3.7 .
• 023B Proof Proof.
• 023C Remark Remark 3.8 .
• 023D Section 4 Asymptotic ansatz and initial error
• 023E Subsection 4.1 Generic region near infinity
• 023F Lemma Lemma 4.1 .
• 023G Subsection 4.2 Asymptotic expansion near D 1
• 023H Subsection 4.3 More background on the Tian-Yau metric
• 023I Theorem Theorem 4.2 .
• 023J Proposition Proposition 4.3 .
• 023K Remark Remark 4.4 .
• 023L Lemma Lemma 4.5 .
• 023M Proof Proof.
• 023N Corollary Corollary 4.6 .
• 023P Subsection 4.4 Non-generic region near infinity
• 023Q Lemma Lemma 4.7 .
• 023R Lemma Lemma 4.8 .
• 023S Lemma Lemma 4.9 .
• 023T Proof Proof.
• 023U Lemma Lemma 4.10 .
• 023V Proof Proof.
• 023W Subsection 4.5 Refined local ansatz in the non-generic region
• 023X Lemma Lemma 4.11 .
• 023Y Proof Proof.
• 023Z Subsection 4.6 Gluing the regions
• 0240 Lemma Lemma 4.12 .
• 0241 Proof Proof.
• 0242 Corollary Corollary 4.13 .
• 0243 Proof Proof.
• 0244 Subsection 4.7 Distance-like function
• 0245 Lemma Lemma 4.14 .
• 0246 Section 5 Weighted Sobolev and Calabi-Yau metric
• 0247 Subsection 5.1 Hein’s package
• 0248 Subsection 5.2 Sufficient condition for weighted Sobolev
• 0249 Subsection 5.3 Assembling the pieces
• 024A Theorem Theorem 5.1 .
• 024B Proof Proof.
• 024C Section Appendix: Hypergeometric functions
• 024D Lemma Lemma 5.2 .
• 024E Proof Proof.
• 024F Proposition Proposition 5.3 .
• 024G Proof Proof.
• 024H Lemma Lemma 5.4 .
• 024I Section 1. Introduction
• 024J Theorem Theorem 1.1 .
• 024K Section 2. Proof of Theorem 1.1
• 024L Proof Proof of Theorem 1.1 .
• 024M Lemma Lemma 2.1 .
• 024N Lemma Lemma 2.2 .
• 024P Proof Proof.
• 024Q Lemma Lemma 2.3 .
• 024R Proof Proof.
• 024S Proof Proof of Lemma 2.1 .
• 024T Example Example 2.4 .
• 024U Section Introduction
• 024V Theorem Theorem 0.1 .
• 024W Section 1. Notation and preliminaries
• 024X Subsection 1.1. Notation
• 024Y Subsubsection 1.1.1.
• 024Z Subsubsection 1.1.2.
• 0250 Subsubsection 1.1.3.
• 0251 Subsubsection 1.1.4.
• 0252 Subsubsection 1.1.5.
• 0253 Subsubsection 1.1.6.
• 0254 Subsubsection 1.1.7.
• 0255 Subsection 1.2. Extension obstruction index
• 0256 Proposition Proposition 1.1 .
• 0257 Proof Proof.
• 0258 Corollary Corollary 1.2 .
• 0259 Proof Proof.
• 025A Subsection 1.3. Normed vector space over a non-archimedean field
• 025B Subsubsection 1.3.1. Orthogonality of norms
• 025C Proposition Proposition 1.3 .
• 025D Proof Proof.
• 025E Lemma Lemma 1.4 .
• 025F Proof Proof.
• 025G Subsubsection 1.3.2. Scalar extension of norms
• 025H Lemma Lemma 1.5 .
• 025I Proof Proof.
• 025J Lemma Lemma 1.6 .
• 025K Proof Proof.
• 025L Corollary Corollary 1.7 .
• 025M Proof Proof.
• 025N Definition Definition 1.8 .
• 025P Proposition Proposition 1.9 .
• 025Q Proof Proof.
• 025R Lemma Lemma 1.10 .
• 025S Proof Proof.
• 025T Lemma Lemma 1.11 .
• 025U Proof Proof.
• 025V Lemma Lemma 1.12 .
• 025W Proof Proof.
• 025X Remark Remark 1.13 .
• 025Y Subsubsection 1.3.3. Lattices and norms
• 025Z Proposition Proposition 1.14 .
• 0260 Proof Proof.
• 0261 Lemma Lemma 1.15 .
• 0262 Proof Proof.
• 0263 Lemma Lemma 1.16 .
• 0264 Proof Proof.
• 0265 Proposition Proposition 1.17 .
• 0266 Proof Proof.
• 0267 Proposition Proposition 1.18 .
• 0268 Proof Proof.
• 0269 Proposition Proposition 1.19 .
• 026A Proof Proof.
• 026B Section 2. Seminorm and integral extension
• 026C Theorem Theorem 2.1 .
• 026D Proof Proof.
• 026E Corollary Corollary 2.2 .
• 026F Proof Proof.
• 026G Lemma Lemma 2.3 .
• 026H Proof Proof.
• 026I Section 3. Continuous metrics of invertible sheaves
• 026J Lemma Lemma 3.1 .
• 026K Proof Proof.
• 026L Subsection 3.1. Extension theorem for a metric arising from a model
• 026M Theorem Theorem 3.2 .
• 026N Proof Proof.
• 026P Claim Claim 3.2.1 .
• 026Q Proof Proof.
• 026R Subsection 3.2. Quotient metric
• 026S Lemma Lemma 3.3 .
• 026T Proof Proof.
• 026U Claim Claim 3.3.1 .
• 026V Proof Proof.
• 026W Corollary Corollary 3.4 .
• 026X Proof Proof.
• 026Y Lemma Lemma 3.5 .
• 026Z Proof Proof.
• 0270 Proposition Proposition 3.6 .
• 0271 Proof Proof.
• 0272 Lemma Lemma 3.7 .
• 0273 Proof Proof.
• 0274 Proposition Proposition 3.8 .
• 0275 Proof Proof.
• 0276 Remark Remark 3.9 .
• 0277 Subsection 3.3. Semipositive metric
• 0278 Proposition Proposition 3.10 .
• 0279 Proof Proof.
• 027A Corollary Corollary 3.11 .
• 027B Proof Proof.
• 027C Corollary Corollary 3.12 .
• 027D Proof Proof.
• 027E Subsection 3.4. The functions σ and μ on X an
• 027F Lemma Lemma 3.13 .
• 027G Proof Proof.
• 027H Lemma Lemma 3.14 .
• 027I Proof Proof.
• 027J Lemma Lemma 3.15 .
• 027K Proof Proof.
• 027L Proposition Proposition 3.16 .
• 027M Proposition Proposition 3.17 .
• 027N Proof Proof.
• 027P Remark Remark 3.18 .
• 027Q Section 4. Extension theorem
• 027R Theorem Theorem 4.1 .
• 027S Proof Proof.
• 027T Claim Claim 4.1.1 .
• 027U Proof Proof.
• 027V Theorem Theorem 4.2 .
• 027W Proof Proof.
• 027X Claim Claim 4.2.1 .
• 027Y Proof Proof.
• 027Z Section Introduction
• 0280 Section 1. Proof of Theorem A
• 0281 Proposition Proposition 1.1 .
• 0282 Proof Proof.
• 0283 Proposition Proposition 1.2 .
• 0284 Proof Proof.
• 0285 Proposition Proposition 1.3 .
• 0286 Proof Proof.
• 0287 Proof Proof of Theorem A.
• 0288 Section 2. Extension of qpsh functions
• 0289 Subsection 2.1. The smooth case
• 028A Proposition Proposition 2.1 .
• 028B Proof Proof.
• 028C Subsection 2.2. Proof of Theorem B
• 028D Theorem Theorem 2.2 .
• 028E Lemma Lemma 2.3 .
• 028F Proof Proof.
• 028G Proof Proof of Theorem 2.2
• 028H Section 3. Algebraic subvarieties of ℂ n
• 028I Subsection 3.1. Extension preserving the Lelong class
• 028J Proposition Proposition 3.1 .
• 028K Proof Proof.
• 028L Subsection 3.2. Explicit examples
• 028M Example Example 3.2 .
• 028N Example Example 3.3 .
• 028P Proposition Proposition 3.4 .
• 028Q Proof Proof.
• 028R Section 1 Introduction
• 028S Theorem Theorem 1
• 028T Theorem Theorem 2
• 028U Section 2 General L ∞ estimates for the solutions.
• 028V Theorem Theorem 3
• 028W Lemma Lemma 1
• 028X Lemma Lemma 2
• 028Y Claim Claim 1
• 028Z Claim Claim 2
• 0290 Lemma Lemma 3
• 0291 Claim Claim 3
• 0292 Lemma Lemma 4
• 0293 Lemma Lemma 5
• 0294 Theorem Theorem 4
• 0295 Corollary Corollary 1
• 0296 Claim Claim 4
• 0297 Section 3 The domain of definition of the complex Monge-Ampère operator.
• 0298 Definition Definition 1
• 0299 Theorem Theorem 5
• 029A Claim Claim 5
• 029B Claim Claim 6
• 029C Corollary Corollary 2
• 029D Corollary Corollary 3
• 029E Section 4 Existence and uniqueness of the solutions.
• 029F Theorem Theorem 6
• 029G Lemma Lemma 6
• 029H Lemma Lemma 7
• 029I Section 5 Higher order regularity of the solutions.
• 029J Theorem Theorem 7
• 029K Definition Definition 2
• 029L Definition Definition 3
• 029M Lemma Lemma 8
• 029N Section 6 Appendix
• 029P Section 1 Background
• 029Q Subsection 1.1 Gauge theory and holomorphic bundles.
• 029R Subsection 1.2 Symplectic and complex structures
• 029S Subsection 1.3 The equations
• 029T Section 2 Toric manifolds
• 029U Subsection 2.1 Local differential geometry
• 029V Subsubsection 2.1.1 Complex coordinates
• 029W Subsubsection 2.1.2 Symplectic coordinates
• 029X Subsection 2.2 The global structure
• 029Y Subsubsection 2.2.1 Complex charts
• 029Z Subsubsection 2.2.2 Symplectic construction
• 02A0 Subsubsection 2.2.3 Algebraic construction
• 02A1 Subsubsection 2.2.4 Real forms
• 02A2 Subsection 2.3 Algebraic metrics and asymptotics
• 02A3 Subsubsection 2.3.1 Asymptotics of L 2 -metrics
• 02A4 Subsubsection 2.3.2 The Veronese embedding and the Central Limit theorem
• 02A5 Subsection 2.4 Extremal metrics on toric varieties
• 02A6 Conjecture Conjecture 1
• 02A7 Section 3 Toric Fano manifolds
• 02A8 Subsection 3.1 The Kahler-Ricci soliton equation
• 02A9 Theorem Theorem 1
• 02AA Theorem Theorem 2
• 02AB Subsection 3.2 Continuity method, convexity and a fundamental inequality
• 02AC Subsection 3.3 A priori estimate
• 02AD Subsection 3.4 The method of Wang and Zhu
• 02AE Proposition Proposition 1
• 02AF Proposition Proposition 2
• 02AG Section 4 Variants of toric differential geometry
• 02AH Subsection 4.1 Multiplicity-free manifolds
• 02AI Subsection 4.2 Manifolds with a dense orbit
• 02AJ Lemma Lemma 1
• 02AK Section 5 The Mukai-Umemura manifold and its deformations
• 02AL Subsection 5.1 Mukai’s construction
• 02AM Subsection 5.2 Topological and symplectic picture
• 02AN Subsection 5.3 Deformations
• 02AP Subsection 5.4 The α -invariant
• 02AQ Theorem Theorem 3
• 02AR Lemma Lemma 2
• 02AS Lemma Lemma 3
• 02AT Section 1. Introduction
• 02AU Theorem Theorem 1.1 .
• 02AV Theorem Theorem 1.2 .
• 02AW Section 2. Background
• 02AX Subsection 2.1. Convergence Theory
• 02AY Subsection 2.2. Complex differential geometry: the Hormänder technique
• 02AZ Proposition Proposition 2.1 .
• 02B0 Lemma Lemma 2.2 .
• 02B1 Proposition Proposition 2.3 .
• 02B2 Proposition Proposition 2.4 .
• 02B3 Remark Remark 2.5 .
• 02B4 Section 3. Proof of Theorem 1.1
• 02B5 Subsection 3.1. Reduction to the local case
• 02B6 Lemma Lemma 3.1 .
• 02B7 Theorem Theorem 3.2 .
• 02B8 Proposition Proposition 3.3 .
• 02B9 Lemma Lemma 3.4 .
• 02BA Subsection 3.2. Proof of Theorem 3.2
• 02BB Subsubsection 3.2.1. Cut-offs
• 02BC Proposition Proposition 3.5 .
• 02BD Lemma Lemma 3.6 .
• 02BE Subsubsection 3.2.2. The topological obstruction
• 02BF Proposition Proposition 3.7 .
• 02BG Proposition Proposition 3.8 .
• 02BH Proposition Proposition 3.9 .
• 02BI Proposition Proposition 3.10 .
• 02BJ Proposition Proposition 3.11 .
• 02BK Subsubsection 3.2.3. Completion of Proof
• 02BL Proposition Proposition 3.12 .
• 02BM Proposition Proposition 3.13 .
• 02BN Section 4. Connections with algebraic geometry
• 02BP Subsection 4.1. Proof of Theorem 2
• 02BQ Lemma Lemma 4.1 .
• 02BR Lemma Lemma 4.2 .
• 02BS Lemma Lemma 4.3 .
• 02BT Subsection 4.2. More analysis
• 02BU Lemma Lemma 4.4 .
• 02BV Lemma Lemma 4.5 .
• 02BW Proposition Proposition 4.6 .
• 02BX Proposition Proposition 4.7 .
• 02BY Subsection 4.3. Recap
• 02BZ Subsubsection 4.3.1. Completion of proof of Theorem 1.2
• 02C0 Lemma Lemma 4.8 .
• 02C1 Lemma Lemma 4.9 .
• 02C2 Proposition Proposition 4.10 .
• 02C3 Lemma Lemma 4.11 .
• 02C4 Lemma Lemma 4.12 .
• 02C5 Lemma Lemma 4.13 .
• 02C6 Subsection 4.4. Further results
• 02C7 Proposition Proposition 4.14 .
• 02C8 Proof Proof.
• 02C9 Proposition Proposition 4.15 .
• 02CA Proof Proof.
• 02CB Remark Remark 4.16 .
• 02CC Section 5. Structure of three dimensional tangent cones
• 02CD Theorem Theorem 5.1 .
• 02CE Lemma Lemma 5.2 .
• 02CF Proof Proof.
• 02CG Lemma Lemma 5.3 .
• 02CH Proof Proof.
• 02CI Proposition Proposition 5.4 .
• 02CJ Proof Proof.
• 02CK Lemma Lemma 5.5 .
• 02CL Proof Proof.
• 02CM Proposition Proposition 5.6 .
• 02CN Proof Proof.
• 02CP Theorem Theorem 5.7 .
• 02CQ Lemma Lemma 5.8 .
• 02CR Proof Proof.
• 02CS Lemma Lemma 5.9 .
• 02CT Proof Proof.
• 02CU Proposition Proposition 5.10 .
• 02CV Proof Proof.
• 02CW Lemma Lemma 5.11 .
• 02CX Proof Proof.
• 02CY Proposition Proposition 5.12 .
• 02CZ Proof Proof.
• 02D0 Lemma Lemma 5.13 .
• 02D1 Proposition Proposition 5.14 .
• 02D2 Proof Proof.
• 02D3 Subsection 5.1. Further discussion
• 02D4 Conjecture Conjecture 5.15 .
• 02D5 Section Introduction
• 02D6 Section Notations and organization of the paper
• 02D7 Section 1. Weak solutions to Monge-Ampère equations
• 02D8 Definition Definition 1.1 .
• 02D9 Proposition Proposition 1.2 .
• 02DA Proof Proof.
• 02DB Definition Definition 1.3 .
• 02DC Proposition Proposition 1.4 .
• 02DD Proof Proof.
• 02DE Remark Remark 1.5 .
• 02DF Section 2. Continuous solutions
• 02DG Theorem Theorem 2.1 .
• 02DH Lemma Lemma 2.2 .
• 02DI Proof Proof.
• 02DJ Lemma Lemma 2.3 .
• 02DK Proof Proof.
• 02DL Remark Remarks 2.4 .
• 02DM Proposition Proposition 2.5 .
• 02DN Proof Proof.
• 02DP Section 3. More regularity
• 02DQ Subsection 3.1. Measures with density
• 02DR Proposition Proposition 3.1 .
• 02DS Proof Proof.
• 02DT Lemma Lemma 3.2 .
• 02DU Proof Proof.
• 02DV Subsection 3.2. Hölder continuity
• 02DW Proposition Proposition 3.3 .
• 02DX Proof Proof.
• 02DY Theorem Theorem 3.4 .
• 02DZ Proof Proof.
• 02E0 Subsection 3.3. Regularity on the smooth locus
• 02E1 Theorem Theorem 3.5 .
• 02E2 Remark Remark 3.6 .
• 02E3 Subsubsection Preliminary considerations
• 02E4 Subsubsection The simplest case
• 02E5 Subsubsection Case where ∩ i { t i = 0 } = ∅
• 02E6 Remark Remark 3.7 .
• 02E7 Subsection 3.4. Formal reduction of the general case to the second case using smooth orbifolds
• 02E8 Remark Remarks 3.8 .
• 02E9 Section 4. More Monge-Ampère equations
• 02EA Theorem Theorem 4.1 .
• 02EB Proof Proof.
• 02EC Lemma Lemma 4.2 .
• 02ED Proof Proof.
• 02EE Proposition Proposition 4.3 .
• 02EF Proof Proof.
• 02EG Proposition Proposition 4.4 .
• 02EH Proof Proof.
• 02EI Theorem Theorem 4.5 .
• 02EJ Proof Proof.
• 02EK Remark Remark 4.6 .
• 02EL Section 5. Singularities in Mori theory
• 02EM Subsection 5.1. Log terminal singularities
• 02EN Definition Definition 5.1 .
• 02EP Definition Definition 5.2 .
• 02EQ Definition Definition 5.3 .
• 02ER Theorem Theorem 5.4 .
• 02ES Example Examples 5.5 .
• 02ET Example Examples 5.6 .
• 02EU Subsection 5.2. Normal Kähler spaces
• 02EV Subsubsection Plurisubharmonic functions
• 02EW Subsubsection Semi-Kähler currents
• 02EX Definition Definition 5.7 .
• 02EY Definition Definition 5.8 .
• 02EZ Example Example 5.9 .
• 02F0 Subsubsection Chern-Weil forms and hermitian metrics
• 02F1 Remark Remark 5.10 .
• 02F2 Definition Definition 5.11 .
• 02F3 Proposition Proposition 5.12 .
• 02F4 Proof Proof.
• 02F5 Section 6. Adapted volume forms
• 02F6 Subsection 6.1. Monge-Ampère equations on normal Kähler spaces
• 02F7 Proposition Proposition 6.1 .
• 02F8 Lemma Lemma 6.2 .
• 02F9 Theorem Theorem 6.3 .
• 02FA Proof Proof.
• 02FB Subsection 6.2. Adapted measures on log terminal Kähler spaces
• 02FC Lemma Lemma 6.4 .
• 02FD Proof Proof.
• 02FE Definition Definition 6.5 .
• 02FF Remark Remark 6.6 .
• 02FG Subsection 6.3. Adapted volume forms for klt Kähler pairs
• 02FH Definition Definition 6.7 .
• 02FI Lemma Lemma 6.8 .
• 02FJ Section 7. Singular Kähler-Einstein metrics
• 02FK Subsection 7.1. Singular Ricci curvature
• 02FL Subsubsection The smooth case
• 02FM Lemma Lemma 7.1 .
• 02FN Subsubsection Adapted measures and hermitian metrics on the canonical sheaf
• 02FP Definition Definition 7.2 .
• 02FQ Definition Definition 7.3 .
• 02FR Subsection 7.2. Singular Ricci flat metrics
• 02FS Definition Definition 7.4 .
• 02FT Theorem Theorem 7.5 .
• 02FU Corollary Corollary 7.6 .
• 02FV Proof Proof.
• 02FW Example Example 7.7 .
• 02FX Subsection 7.3. Singular Kähler-Einstein metrics of negative curvature
• 02FY Theorem Theorem 7.8 .
• 02FZ Proof Proof.
• 02G0 Remark Remark 7.9 .
• 02G1 Subsubsection Connection with [Ts]
• 02G2 Example Example 7.10 .
• 02G3 Subsection 7.4. Singular KE metrics on klt pairs
• 02G4 Definition Definition 7.11 .
• 02G5 Theorem Theorem 7.12 .
• 02G6 Proof Proof.
• 02G7 Section 1 Introduction
• 02G8 Section 2 Proof of the theorem
• 02G9 Subsection 2.1 Preliminary remarks
• 02GA Lemma Lemma 2.1
• 02GB Subsection 2.2 Uniform domination by capacity
• 02GC Lemma Lemma 2.2
• 02GD Subsection 2.3 Uniform normalization
• 02GE Paragraph Affinoid domains and structural algebra
• 02GF Paragraph Special domains and acyclicity of structural presheaf
• 02GG Section 1. Introduction
• 02GH Theorem Theorem 1.1 .
• 02GI Remark Remark .
• 02GJ Remark Remark .
• 02GK Subsection Plan of the paper
• 02GL Subsection Acknowledgements
• 02GM Section 2. Definite triples and hyperkähler structures on 4 –manifolds
• 02GN Definition Definition 2.1 .
• 02GP Definition Definition 2.4 .
• 02GQ Subsection 2.1. The deformation problem
• 02GR Remark Remark .
• 02GS Remark Remark .
• 02GT Section 3. ALF gravitational instantons
• 02GU Definition Definition 3.1 .
• 02GV Subsection 3.1. The Gibbons–Hawking ansatz
• 02GW Subsection 3.2. Families of ALF gravitational instantons
• 02GX Definition Definition 3.6 .
• 02GY Remark Remark .
• 02GZ Subsubsection 3.2.1. ALF spaces of cyclic type
• 02H0 Subsubsection 3.2.2. ALF spaces of dihedral type
• 02H1 Remark Remark .
• 02H2 Remark Remark .
• 02H3 Remark Remark 3.7 .
• 02H4 Section 4. Gibbons–Hawking ansatz on a punctured 3 –torus
• 02H5 Subsection 4.1. Dirac monopoles on a punctured torus
• 02H6 Proposition Proposition 4.3 .
• 02H7 Proof Proof.
• 02H8 Subsection 4.2. Collapsed S 1 –invariant hyperkähler metrics
• 02H9 Lemma Lemma 4.7 .
• 02HA Remark Remark .
• 02HB Remark Remark 4.8 .
• 02HC Lemma Lemma 4.9 .
• 02HD Proof Proof.
• 02HE Lemma Lemma 4.10 .
• 02HF Proof Proof.
• 02HG Section 5. Approximate hyperkähler metrics
• 02HH Subsection 5.1. The 4 –manifold M ϵ
• 02HI Proposition Proposition 5.1 .
• 02HJ Proof Proof.
• 02HK Remark Remark .
• 02HL Subsection 5.2. The definite triple ω ¯ ϵ
• 02HM Remark Remark .
• 02HN Lemma Lemma 5.4 .
• 02HP Proof Proof.
• 02HQ Remark Remark .
• 02HR Subsubsection 5.2.1. The error
• 02HS Section 6. Perturbation to hyperkähler metrics
• 02HT Subsection 6.1. The linear operator d ∗ + 2 ​ d + for collapsing Gibbons–Hawking metrics
• 02HU Remark Remark .
• 02HV Subsection 6.2. Weighted Hölder spaces
• 02HW Definition Definition 6.8 .
• 02HX Lemma Lemma 6.9 .
• 02HY Proof Proof.
• 02HZ Proposition Proposition 6.10 .
• 02I0 Proof Proof.
• 02I1 Subsection 6.3. The linear estimate
• 02I2 Proposition Proposition 6.11 .
• 02I3 Proof Proof.
• 02I4 Subsection 6.4. The non-linear problem
• 02I5 Lemma Lemma 6.13 .
• 02I6 Lemma Lemma 6.14 .
• 02I7 Proof Proof.
• 02I8 Theorem Theorem 6.15 .
• 02I9 Proof Proof.
• 02IA Section 7. Stable minimal surfaces
• 02IB Theorem Theorem 7.1 .
• 02IC Proof Proof.
• 02ID Section 1. Introduction
• 02IE Theorem Theorem 1.2 .
• 02IF Section 2. Metrized line bundles and their associated heights
• 02IG Subsection 2.1. Smooth metrics in the Archimedean case
• 02IH Definition Definition 2.1 .
• 02II Example Example 2.2 .
• 02IJ Definition Definition 2.3 .
• 02IK Example Example 2.4 .
• 02IL Remark Remark 2.5 .
• 02IM Subsection 2.2. Berkovich spaces of schemes
• 02IN Definition Definition 2.6 .
• 02IP Theorem Theorem 2.7 .
• 02IQ Proof Proof.
• 02IR Example Example 2.8 .
• 02IS Remark Remark 2.9 .
• 02IT Subsection 2.3. Algebraic metrics in the non-Archimedean case
• 02IU Definition Definition 2.10 .
• 02IV Definition Definition 2.11 .
• 02IW Definition Definition 2.16 .
• 02IX Definition Definition 2.17 .
• 02IY Proposition Proposition 2.18 .
• 02IZ Proof Proof.
• 02J0 Proposition Proposition 2.19 .
• 02J1 Proof Proof.
• 02J2 Proposition Proposition 2.21 .
• 02J3 Proof Proof.
• 02J4 Proposition Proposition 2.23 .
• 02J5 Proof Proof.
• 02J6 Example Example 2.24 .
• 02J7 Example Example 2.25 .
• 02J8 Definition Definition 2.26 .
• 02J9 Example Example 2.27 .
• 02JA Definition Definition 2.28 .
• 02JB Remark Remark 2.30 .
• 02JC Subsection 2.4. Approachable and integrable metrics, measures and local heights
• 02JD Definition Definition 2.31 .
• 02JE Example Example 2.32 .
• 02JF Proposition Proposition 2.33 .
• 02JG Proof Proof.
• 02JH Definition Definition 2.34 .
• 02JI Proposition Proposition 2.35 .
• 02JJ Proof Proof.
• 02JK Proposition Proposition 2.36 .
• 02JL Proof Proof.
• 02JM Proposition Proposition 2.37 .
• 02JN Proof Proof.
• 02JP Definition Definition 2.38 .
• 02JQ Definition Definition 2.39 .
• 02JR Remark Remark 2.41 .
• 02JS Remark Remark 2.42 .
• 02JT Remark Remark 2.44 .
• 02JU Remark Remark 2.45 .
• 02JV Theorem Theorem 2.46 .
• 02JW Proof Proof.
• 02JX Subsection 2.5. Adelic metrics and global heights
• 02JY Definition Definition 2.47 .
• 02JZ Definition Definition 2.48 .
• 02K0 Example Example 2.49 .
• 02K1 Example Example 2.50 .
• 02K2 Definition Definition 2.51 .
• 02K3 Definition Definition 2.52 .
• 02K4 Definition Definition 2.53 .
• 02K5 Proposition Proposition 2.55 .
• 02K6 Proof Proof.
• 02K7 Definition Definition 2.56 .
• 02K8 Theorem Theorem 2.57 .
• 02K9 Proof Proof.
• 02KA Section 3. The Legendre-Fenchel duality
• 02KB Subsection 3.1. Convex sets and convex decompositions
• 02KC Definition Definition 3.1 .
• 02KD Definition Definition 3.2 .
• 02KE Definition Definition 3.3 .
• 02KF Definition Definition 3.6 .
• 02KG Definition Definition 3.7 .
• 02KH Example Example 3.8 .
• 02KI Proposition Proposition 3.9 .
• 02KJ Proof Proof.
• 02KK Definition Definition 3.10 .
• 02KL Lemma Lemma 3.11 .
• 02KM Proof Proof.
• 02KN Definition Definition 3.12 .
• 02KP Definition Definition 3.13 .
• 02KQ Remark Remark 3.14 .
• 02KR Corollary Corollary 3.15 .
• 02KS Proof Proof.
• 02KT Subsection 3.2. The Legendre-Fenchel dual of a concave function
• 02KU Example Example 3.16 .
• 02KV Proposition Proposition 3.17 .
• 02KW Proof Proof.
• 02KX Proposition Proposition 3.18 .
• 02KY Proof Proof.
• 02KZ Proposition Proposition 3.21 .
• 02L0 Proof Proof.
• 02L1 Definition Definition 3.23 .
• 02L2 Lemma Lemma 3.24 .
• 02L3 Proof Proof.
• 02L4 Proposition Proposition 3.26 .
• 02L5 Proof Proof.
• 02L6 Proposition Proposition 3.27 .
• 02L7 Proof Proof.
• 02L8 Lemma Lemma 3.28 .
• 02L9 Proof Proof.
• 02LA Definition Definition 3.31 .
• 02LB Definition Definition 3.32 .
• 02LC Theorem Theorem 3.33 .
• 02LD Proof Proof.
• 02LE Definition Definition 3.34 .
• 02LF Example Example 3.36 .
• 02LG Example Example 3.37 .
• 02LH Subsection 3.3. Operations on concave functions and duality
• 02LI Proposition Proposition 3.38 .
• 02LJ Proof Proof.
• 02LK Remark Remark 3.39 .
• 02LL Proposition Proposition 3.40 .
• 02LM Proof Proof.
• 02LN Lemma Lemma 3.41 .
• 02LP Proof Proof.
• 02LQ Definition Definition 3.42 .
• 02LR Proposition Proposition 3.43 .
• 02LS Proof Proof.
• 02LT Proposition Proposition 3.45 .
• 02LU Proof Proof.
• 02LV Proposition Proposition 3.46 .
• 02LW Proof Proof.
• 02LX Definition Definition 3.48 .
• 02LY Proposition Proposition 3.50 .
• 02LZ Proof Proof.
• 02M0 Subsection 3.4. The differentiable case
• 02M1 Definition Definition 3.51 .
• 02M2 Theorem Theorem 3.52 .
• 02M3 Proof Proof.
• 02M4 Example Example 3.53 .
• 02M5 Proposition Proposition 3.55 .
• 02M6 Proof Proof.
• 02M7 Example Example 3.57 .
• 02M8 Subsection 3.5. The piecewise affine case
• 02M9 Definition Definition 3.58 .
• 02MA Lemma Lemma 3.59 .
• 02MB Proof Proof.
• 02MC Definition Definition 3.60 .
• 02MD Proposition Proposition 3.64 .
• 02ME Proof Proof.
• 02MF Example Example 3.65 .
• 02MG Definition Definition 3.66 .
• 02MH Definition Definition 3.67 .
• 02MI Definition Definition 3.68 .
• 02MJ Proposition Proposition 3.69 .
• 02MK Proof Proof.
• 02ML Example Example 3.70 .
• 02MM Example Example 3.71 .
• 02MN Proposition Proposition 3.72 .
• 02MP Proof Proof.
• 02MQ Definition Definition 3.73 .
• 02MR Remark Remark 3.74 .
• 02MS Proposition Proposition 3.75 .
• 02MT Proof Proof.
• 02MU Example Example 3.76 .
• 02MV Proposition Proposition 3.77 .
• 02MW Proof Proof.
• 02MX Proposition Proposition 3.78 .
• 02MY Proof Proof.
• 02MZ Lemma Lemma 3.79 .
• 02N0 Proof Proof.
• 02N1 Proposition Proposition 3.80 .
• 02N2 Proof Proof.
• 02N3 Proposition Proposition 3.81 .
• 02N4 Proof Proof.
• 02N5 Subsection 3.6. Differences of concave functions
• 02N6 Definition Definition 3.82 .
• 02N7 Proposition Proposition 3.83 .
• 02N8 Proof Proof.
• 02N9 Corollary Corollary 3.84 .
• 02NA Proof Proof.
• 02NB Definition Definition 3.85 .
• 02NC Proposition Proposition 3.87 .
• 02ND Proof Proof.
• 02NE Definition Definition 3.88 .
• 02NF Proposition Proposition 3.89 .
• 02NG Proof Proof.
• 02NH Definition Definition 3.90 .
• 02NI Proposition Proposition 3.91 .
• 02NJ Proof Proof.
• 02NK Subsection 3.7. Monge-Ampère measures
• 02NL Definition Definition 3.92 .
• 02NM Proposition Proposition 3.93 .
• 02NN Proof Proof.
• 02NP Proposition Proposition 3.94 .
• 02NQ Proof Proof.
• 02NR Proposition Proposition 3.95 .
• 02NS Proof Proof.
• 02NT Example Example 3.96 .
• 02NU Theorem Theorem 3.97 .
• 02NV Proof Proof.
• 02NW Definition Definition 3.102 .
• 02NX Notation Notation 3.103 .
• 02NY Corollary Corollary 3.104 .
• 02NZ Proof Proof.
• 02P0 Example Example 3.106 .
• 02P1 Definition Definition 3.107 .
• 02P2 Proposition Proposition 3.108 .
• 02P3 Proof Proof.
• 02P4 Definition Definition 3.109 .
• 02P5 Proposition Proposition 3.111 .
• 02P6 Proof Proof.
• 02P7 Corollary Corollary 3.112 .
• 02P8 Proof Proof.
• 02P9 Definition Definition 3.113 .
• 02PA Section 4. Toric varieties
• 02PB Subsection 4.1. Fans and toric varieties
• 02PC Definition Definition 4.1 .
• 02PD Theorem Theorem 4.2 .
• 02PE Proof Proof.
• 02PF Example Example 4.3 .
• 02PG Subsection 4.2. Orbits and equivariant morphisms
• 02PH Proposition Proposition 4.6 .
• 02PI Proof Proof.
• 02PJ Definition Definition 4.7 .
• 02PK Theorem Theorem 4.9 .
• 02PL Proof Proof.
• 02PM Example Example 4.10 .
• 02PN Definition Definition 4.12 .
• 02PP Example Example 4.13 .
• 02PQ Subsection 4.3. 𝕋 -Cartier divisors and toric line bundles
• 02PR Definition Definition 4.14 .
• 02PS Definition Definition 4.15 .
• 02PT Theorem Theorem 4.18 .
• 02PU Proof Proof.
• 02PV Definition Definition 4.19 .
• 02PW Remark Remark 4.20 .
• 02PX Remark Remark 4.21 .
• 02PY Theorem Theorem 4.22 .
• 02PZ Proof Proof.
• 02Q0 Notation Notation 4.23 .
• 02Q1 Definition Definition 4.24 .
• 02Q2 Example Example 4.26 .
• 02Q3 Theorem Theorem 4.27 .
• 02Q4 Proof Proof.
• 02Q5 Remark Remark 4.28 .
• 02Q6 Corollary Corollary 4.29 .
• 02Q7 Proof Proof.
• 02Q8 Proposition Proposition 4.31 .
• 02Q9 Proof Proof.
• 02QA Example Example 4.32 .
• 02QB Proposition Proposition 4.34 .
• 02QC Proposition Proposition 4.35 .
• 02QD Remark Remark 4.36 .
• 02QE Subsection 4.4. Positivity properties of 𝕋 -Cartier divisors
• 02QF Proposition Proposition 4.37 .
• 02QG Proof Proof.
• 02QH Remark Remark 4.40 .
• 02QI Definition Definition 4.41 .
• 02QJ Theorem Theorem 4.42 .
• 02QK Proof Proof.
• 02QL Remark Remark 4.43 .
• 02QM Example Example 4.44 .
• 02QN Proposition Proposition 4.45 .
• 02QP Proof Proof.
• 02QQ Proposition Proposition 4.46 .
• 02QR Proof Proof.
• 02QS Proposition Proposition 4.47 .
• 02QT Proof Proof.
• 02QU Corollary Corollary 4.52 .
• 02QV Proof Proof.
• 02QW Example Example 4.53 .
• 02QX Theorem Theorem 4.54 .
• 02QY Proof Proof.
• 02QZ Subsection 4.5. Toric schemes over a discrete valuation ring
• 02R0 Definition Definition 4.55 .
• 02R1 Definition Definition 4.56 .
• 02R2 Theorem Theorem 4.59 .
• 02R3 Proof Proof.
• 02R4 Theorem Theorem 4.60 .
• 02R5 Proof Proof.
• 02R6 Example Example 4.62 .
• 02R7 Definition Definition 4.63 .
• 02R8 Proposition Proposition 4.64 .
• 02R9 Proof Proof.
• 02RA Proposition Proposition 4.65 .
• 02RB Proof Proof.
• 02RC Remark Remark 4.66 .
• 02RD Remark Remark 4.67 .
• 02RE Definition Definition 4.68 .
• 02RF Lemma Lemma 4.69 .
• 02RG Proof Proof.
• 02RH Definition Definition 4.70 .
• 02RI Definition Definition 4.71 .
• 02RJ Proposition Proposition 4.72 .
• 02RK Proof Proof.
• 02RL Subsection 4.6. 𝕋 -Cartier divisors on toric schemes
• 02RM Definition Definition 4.74 .
• 02RN Example Example 4.75 .
• 02RP Definition Definition 4.76 .
• 02RQ Definition Definition 4.77 .
• 02RR Proposition Proposition 4.78 .
• 02RS Proposition Proposition-Definition 4.79 .
• 02RT Proof Proof.
• 02RU Theorem Theorem 4.80 .
• 02RV Proof Proof.
• 02RW Theorem Theorem 4.81 .
• 02RX Proof Proof.
• 02RY Remark Remark 4.83 .
• 02RZ Proposition Proposition 4.84 .
• 02S0 Proof Proof.
• 02S1 Example Example 4.86 .
• 02S2 Proposition Proposition 4.89 .
• 02S3 Proof Proof.
• 02S4 Proposition Proposition 4.91 .
• 02S5 Proof Proof.
• 02S6 Example Example 4.92 .
• 02S7 Proposition Proposition 4.94 .
• 02S8 Subsection 4.7. Positivity on toric schemes
• 02S9 Theorem Theorem 4.95 .
• 02SA Proof Proof.
• 02SB Definition Definition 4.96 .
• 02SC Theorem Theorem 4.97 .
• 02SD Proof Proof.
• 02SE Corollary Corollary 4.98 .
• 02SF Proof Proof.
• 02SG Proposition Proposition 4.99 .
• 02SH Proof Proof.
• 02SI Lemma Lemma 4.102 .
• 02SJ Proof Proof.
• 02SK Proposition Proposition 4.103 .
• 02SL Proof Proof.
• 02SM Proposition Proposition 4.105 .
• 02SN Proof Proof.
• 02SP Remark Remark 4.107 .
• 02SQ Proposition Proposition 4.108 .
• 02SR Proof Proof.
• 02SS Example Example 4.109 .
• 02ST Section 5. Metrics and measures on toric varieties
• 02SU Subsection 5.1. The variety with corners X Σ ​ ( ℝ ≥ 0 )
• 02SV Lemma Lemma 5.1 .
• 02SW Proof Proof.
• 02SX Proposition Proposition-Definition 5.2 .
• 02SY Proof Proof.
• 02SZ Remark Remark 5.8 .
• 02T0 Proposition Proposition 5.9 .
• 02T1 Proposition Proposition 5.10 .
• 02T2 Subsection 5.2. Toric metrics
• 02T3 Proposition Proposition 5.11 .
• 02T4 Proof Proof.
• 02T5 Definition Definition 5.12 .
• 02T6 Definition Definition 5.14 .
• 02T7 Proposition Proposition 5.16 .
• 02T8 Proof Proof.
• 02T9 Corollary Corollary 5.17 .
• 02TA Proof Proof.
• 02TB Example Example 5.18 .
• 02TC Proposition Proposition 5.19 .
• 02TD Proof Proof.
• 02TE Proposition Proposition-Definition 5.20 .
• 02TF Proof Proof.
• 02TG Proposition Proposition 5.21 .
• 02TH Proof Proof.
• 02TI Proposition Proposition 5.22 .
• 02TJ Proof Proof.
• 02TK Corollary Corollary 5.23 .
• 02TL Proof Proof.
• 02TM Proposition Proposition 5.24 .
• 02TN Corollary Corollary 5.25 .
• 02TP Example Example 5.26 .
• 02TQ Proposition Proposition 5.27 .
• 02TR Proof Proof.
• 02TS Corollary Corollary 5.28 .
• 02TT Proof Proof.
• 02TU Subsection 5.3. Smooth metrics and their associated measures
• 02TV Proposition Proposition 5.29 .
• 02TW Proof Proof.
• 02TX Definition Definition 5.32 .
• 02TY Theorem Theorem 5.33 .
• 02TZ Proof Proof.
• 02U0 Proposition Proposition 5.38 .
• 02U1 Proof Proof.
• 02U2 Subsection 5.4. Algebraic metrics from toric models
• 02U3 Lemma Lemma 5.39 .
• 02U4 Proof Proof.
• 02U5 Corollary Corollary 5.40 .
• 02U6 Proof Proof.
• 02U7 Proposition Proposition 5.41 .
• 02U8 Proof Proof.
• 02U9 Example Example 5.42 .
• 02UA Corollary Corollary 5.43 .
• 02UB Proof Proof.
• 02UC Example Example 5.44 .
• 02UD Question Question 5.45 .
• 02UE Remark Remark 5.46 .
• 02UF Corollary Corollary 5.47 .
• 02UG Proof Proof.
• 02UH Lemma Lemma 5.48 .
• 02UI Proof Proof.
• 02UJ Theorem Theorem 5.49 .
• 02UK Proof Proof.
• 02UL Proposition Proposition 5.51 .
• 02UM Proof Proof.
• 02UN Proposition Proposition 5.52 .
• 02UP Proof Proof.
• 02UQ Proposition Proposition 5.53 .
• 02UR Proof Proof.
• 02US Subsection 5.5. The one-dimensional case
• 02UT Definition Definition 5.54 .
• 02UU Proposition Proposition 5.55 .
• 02UV Proof Proof.
• 02UW Lemma Lemma 5.56 .
• 02UX Proof Proof.
• 02UY Lemma Lemma 5.57 .
• 02UZ Proof Proof.
• 02V0 Lemma Lemma 5.58 .
• 02V1 Proof Proof.
• 02V2 Lemma Lemma 5.60 .
• 02V3 Proof Proof.
• 02V4 Proposition Proposition 5.61 .
• 02V5 Proof Proof.
• 02V6 Proposition Proposition 5.63 .
• 02V7 Proof Proof.
• 02V8 Corollary Corollary 5.65 .
• 02V9 Proof Proof.
• 02VA Corollary Corollary 5.66 .
• 02VB Proof Proof.
• 02VC Subsection 5.6. Algebraic metrics and their associated measures
• 02VD Proposition Proposition 5.67 .
• 02VE Proof Proof.
• 02VF Corollary Corollary 5.68 .
• 02VG Proof Proof.
• 02VH Corollary Corollary 5.69 .
• 02VI Theorem Theorem 5.70 .
• 02VJ Proof Proof.
• 02VK Subsection 5.7. Approachable and integrable metrics
• 02VL Theorem Theorem 5.73 .
• 02VM Proof Proof.
• 02VN Remark Remark 5.74 .
• 02VP Proposition Proposition 5.75 .
• 02VQ Proof Proof.
• 02VR Proposition Proposition 5.80 .
• 02VS Proof Proof.
• 02VT Theorem Theorem 5.81 .
• 02VU Proof Proof.
• 02VV Corollary Corollary 5.83 .
• 02VW Subsection 5.8. Adelic toric metrics
• 02VX Definition Definition 5.84 .
• 02VY Theorem Theorem 5.85 .
• 02VZ Proof Proof.
• 02W0 Corollary Corollary 5.86 .
• 02W1 Proof Proof.
• 02W2 Section 6. Height of toric varieties
• 02W3 Subsection 6.1. Local heights of toric varieties
• 02W4 Definition Definition 6.1 .
• 02W5 Remark Remark 6.3 .
• 02W6 Proposition Proposition 6.4 .
• 02W7 Proof Proof.
• 02W8 Theorem Theorem 6.6 .
• 02W9 Definition Definition 6.8 .
• 02WA Proof Proof of Theorem 6.6 .
• 02WB Remark Remark 6.16 .
• 02WC Corollary Corollary 6.17 .
• 02WD Proof Proof.
• 02WE Corollary Corollary 6.21 .
• 02WF Proof Proof.
• 02WG Remark Remark 6.23 .
• 02WH Proposition Proposition 6.24 .
• 02WI Proof Proof.
• 02WJ Proposition Proposition 6.25 .
• 02WK Proof Proof.
• 02WL Proposition Proposition 6.27 .
• 02WM Proof Proof.
• 02WN Corollary Corollary 6.30 .
• 02WP Example Example 6.31 .
• 02WQ Subsection 6.2. Global heights of toric varieties
• 02WR Definition Definition 6.32 .
• 02WS Remark Remark 6.33 .
• 02WT Remark Remark 6.34 .
• 02WU Proposition Proposition 6.35 .
• 02WV Proof Proof.
• 02WW Theorem Theorem 6.37 .
• 02WX Proof Proof.
• 02WY Corollary Corollary 6.39 .
• 02WZ Proof Proof.
• 02X0 Remark Remark 6.40 .
• 02X1 Section 7. Metrics from polytopes
• 02X2 Subsection 7.1. Integration on polytopes
• 02X3 Definition Definition 7.1 .
• 02X4 Proposition Proposition 7.3 .
• 02X5 Proof Proof.
• 02X6 Corollary Corollary 7.7 .
• 02X7 Proof Proof.
• 02X8 Proposition Proposition 7.8 .
• 02X9 Proof Proof.
• 02XA Example Example 7.10 .
• 02XB Corollary Corollary 7.14 .
• 02XC Proof Proof.
• 02XD Proposition Proposition 7.15 .
• 02XE Proof Proof.
• 02XF Lemma Lemma 7.18 .
• 02XG Proof Proof.
• 02XH Corollary Corollary 7.19 .
• 02XI Proof Proof.
• 02XJ Subsection 7.2. Metrics, heights and entropy
• 02XK Lemma Lemma 7.21 .
• 02XL Proof Proof.
• 02XM Definition Definition 7.23 .
• 02XN Example Example 7.24 .
• 02XP Remark Remark 7.26 .
• 02XQ Proposition Proposition 7.27 .
• 02XR Proof Proof.
• 02XS Example Example 7.29 .
• 02XT Example Example 7.30 .
• 02XU Remark Remark 7.33 .
• 02XV Proposition Proposition 7.34 .
• 02XW Proof Proof.
• 02XX Example Example 7.35 .
• 02XY Remark Remark 7.36 .
• 02XZ Section 8. Variations on Fubini-Study metrics
• 02Y0 Subsection 8.1. Height of toric projective curves
• 02Y1 Proposition Proposition 8.1 .
• 02Y2 Proof Proof.
• 02Y3 Lemma Lemma 8.6 .
• 02Y4 Proof Proof.
• 02Y5 Theorem Theorem 8.7 .
• 02Y6 Remark Remark 8.8 .
• 02Y7 Proof Proof.
• 02Y8 Corollary Corollary 8.12 .
• 02Y9 Proof Proof.
• 02YA Corollary Corollary 8.13 .
• 02YB Proof Proof.
• 02YC Corollary Corollary 8.16 .
• 02YD Proof Proof.
• 02YE Subsection 8.2. Height of toric bundles
• 02YF Lemma Lemma 8.17 .
• 02YG Proof Proof.
• 02YH Corollary Corollary 8.19 .
• 02YI Proposition Proposition 8.20 .
• 02YJ Proof Proof.
• 02YK Lemma Lemma 8.22 .
• 02YL Proof Proof.
• 02YM Proposition Proposition 8.26 .
• 02YN Proof Proof.
• 02YP Remark Remark 8.27 .
• 02YQ Section List of symbols
• 02YR Section Introduction.
• 02YS Definition Definition 0.1 .
• 02YT Conjecture Conjecture 0.2 .
• 02YU Section 1. Moduli of special Lagrangian submanifolds
• 02YV Definition Definition 1.1 .
• 02YW Proposition Proposition 1.2 .
• 02YX Proof Proof.
• 02YY Definition Definition 1.3 .
• 02YZ SourceObject Exercise 1.4 .
• 02Z0 Section 2. Semi-flat mirror symmetry
• 02Z1 Definition Definition 2.1 .
• 02Z2 Proposition Proposition 2.2 .
• 02Z3 Proof Proof.
• 02Z4 Proposition Proposition 2.3 .
• 02Z5 Proof Proof.
• 02Z6 Definition Definition 2.4 .
• 02Z7 SourceObject Exercise 2.5 .
• 02Z8 SourceObject Construction 2.6 (The toy mirror symmetry construction) .
• 02Z9 Section 3. Affine manifolds with singularities
• 02ZA Definition Definition 3.1 .
• 02ZB Example Example 3.2 .
• 02ZC Theorem Theorem 3.3 .
• 02ZD Example Example 3.4 .
• 02ZE Section 4. Tropical geometry
• 02ZF Definition Definition 4.1 .
• 02ZG Question Question 4.2 .
• 02ZH Section 5. The problems with the SYZ conjecture, and how to get around them
• 02ZI Example Example 5.1 .
• 02ZJ Definition Definition 5.2 .
• 02ZK Example Example 5.3 .
• 02ZL Conjecture Conjecture 5.4 .
• 02ZM Conjecture Conjecture 5.5 .
• 02ZN Question Question 5.6 (The reconstruction problem, Version I) .
• 02ZP Section 6. Gromov-Hausdorff limits, algebraic degenerations, and mirror symmetry
• 02ZQ SourceObject Exercise 6.1 .
• 02ZR Section 7. Toric degenerations, the intersection complex and its dual
• 02ZS Definition Definition 7.1 .
• 02ZT Example Example 7.2 .
• 02ZU Lemma Lemma 7.3 .
• 02ZV Definition Definition 7.4 .
• 02ZW Example Example 7.5 .
• 02ZX Example Example 7.6 .
• 02ZY Example Example 7.7 .
• 02ZZ Theorem Theorem 7.8 .
• 0300 Conjecture Conjecture 7.9 .
• 0301 Definition Definition 7.10 .
• 0302 Example Example 7.11 .
• 0303 Question Question 7.12 (The reconstruction problem, Version II) .
• 0304 Section 8. Log structures
• 0305 Definition Definition 8.1 .
• 0306 Example Examples 8.2 .
• 0307 SourceObject Exercise 8.3 .
• 0308 Section 9. The A -model and tropical geometry
• 0309 Definition Definition 9.1 .
• 030A Definition Definition 9.2 .
• 030B Theorem Theorem 9.3 .
• 030C Theorem Theorem 9.4 .
• 030D Section 10. The B -model and tropical geometry
• 030E Theorem Theorem 10.1 .
• 030F Section 11. The tropical vertex
• 030G Definition Definition 11.1 .
• 030H Proposition Proposition 11.2 .
• 030I Example Example 11.3 .
• 030J Theorem Theorem 11.4 .
• 030K Example Example 11.5 .
• 030L Section 12. Other recent results and the future
• 030M Section 1. Introduction
• 030N Theorem Theorem 1.1 .
• 030P Theorem Theorem 1.2 .
• 030Q Theorem Theorem 1.3 .
• 030R Section 2. Hyperkähler mirror symmetry
• 030S Assumption Assumptions 2.1 .
• 030T Lemma Lemma 2.2 .
• 030U Proof Proof.
• 030V Conjecture Conjecture 2.3 .
• 030W Section 3. Semi-flat metrics
• 030X Proposition Proposition 3.1 .
• 030Y Proof Proof.
• 030Z Section 4. Estimates and smooth convergence
• 0310 Lemma Lemma 4.1 .
• 0311 Proof Proof.
• 0312 Lemma Lemma 4.2 .
• 0313 Proof Proof.
• 0314 Proposition Proposition 4.3 .
• 0315 Proof Proof.
• 0316 Lemma Lemma 4.4 .
• 0317 Proof Proof.
• 0318 Lemma Lemma 4.5 .
• 0319 Proof Proof.
• 031A Proposition Proposition 4.6 .
• 031B Proof Proof.
• 031C Lemma Lemma 4.7 .
• 031D Proof Proof.
• 031E Proof Proof.
• 031F Remark Remark 4.8 .
• 031G Proposition Proposition 4.9 .
• 031H Proof Proof.
• 031I Remark Remark 4.10 .
• 031J Section 5. Gromov-Hausdorff convergence
• 031K Lemma Lemma 5.1 .
• 031L Proof Proof.
• 031M Lemma Lemma 5.2 .
• 031N Proof Proof.
• 031P Proof Proof of Theorem 1.2 .
• 031Q Remark Remark 5.3 .
• 031R Definition Definition 1.1
• 031S Question Question 1.2
• 031T Proposition Proposition 2.8
• 031U Lemma Lemma 3.1
• 031V Proposition Proposition 3.2
• 031W Proposition Proposition 3.5
• 031X Corollary Corollary 3.7
• 031Y Proposition Proposition 3.8
• 031Z Lemma Lemma 4.1
• 0320 Lemma Lemma 4.3
• 0321 Theorem Theorem 4.4
• 0322 Theorem Theorem 4.5
• 0323 Lemma Lemma 5.1
• 0324 Lemma Lemma 5.2
• 0325 Lemma Lemma 5.3
• 0326 Lemma Lemma 5.4
• 0327 Lemma Lemma 5.5
• 0328 Theorem Theorem 5.6
• 0329 Definition Definition 6.1
• 032A Conjecture Conjecture 6.2
• 032B Theorem Theorem 6.3
• 032C Section Introduction
• 032D Section 1. Quasiplurisubharmonic functions
• 032E Definition Definition 1.1 .
• 032F Example Example 1.2 .
• 032G Proposition Proposition 1.3 .
• 032H Proof Proof.
• 032I Proposition Proposition 1.4 .
• 032J Proof Proof.
• 032K Remark Remark 1.5 .
• 032L Proposition Proposition 1.6 .
• 032M Proof Proof.
• 032N Proposition Proposition 1.7 .
• 032P Proof Proof.
• 032Q Example Example 1.8 .
• 032R Example Example 1.9 .
• 032S Section 2. Monge-Ampère capacity
• 032T Proposition Proposition 2.1 (Chern-Levine-Nirenberg inequalities) .
• 032U Proof Proof.
• 032V Remark Remark 2.2 .
• 032W Corollary Corollary 2.3 .
• 032X Definition Definition 2.4 .
• 032Y Proposition Proposition 2.5 .
• 032Z Proof Proof.
• 0330 Proposition Proposition 2.6 .
• 0331 Proof Proof.
• 0332 Proposition Proposition 2.7 .
• 0333 Proof Proof.
• 0334 Corollary Corollary 2.8 (Quasicontinuity) .
• 0335 Proof Proof.
• 0336 Example Example 2.9 .
• 0337 Proposition Proposition 2.10 .
• 0338 Proof Proof.
• 0339 Corollary Corollary 2.11 .
• 033A Theorem Theorem 2.12 (Dirichlet Problem) .
• 033B Section 3. Alexander capacity
• 033C Subsection 3.1. Global extremal functions
• 033D Definition Definition 3.1 .
• 033E Theorem Theorem 3.2 .
• 033F Proof Proof.
• 033G Corollary Corollary 3.3 .
• 033H Proof Proof.
• 033I Proposition Proposition 3.4 .
• 033J Proof Proof.
• 033K Example Example 3.5 .
• 033L Proposition Proposition 3.6 .
• 033M Proof Proof.
• 033N Subsection 3.2. Alexander capacity
• 033P Definition Definition 3.7 .
• 033Q Proposition Proposition 3.8 .
• 033R Proof Proof.
• 033S Proposition Proposition 3.9 .
• 033T Remark Remark 3.10 .
• 033U Example Example 3.11 .
• 033V Example Example 3.12 .
• 033W Remark Remark 3.13 .
• 033X Section 4. Tchebychev constants
• 033Y Proposition Proposition 4.1 .
• 033Z Proof Proof.
• 0340 Theorem Theorem 4.2 .
• 0341 Proof Proof.
• 0342 Theorem Theorem 4.3 .
• 0343 Proof Proof.
• 0344 Remark Remark 4.4 .
• 0345 Section 5. Comparison of capacities and applications
• 0346 Subsection 5.1. Josefson’s theorem
• 0347 Proposition Proposition 5.1 .
• 0348 Proof Proof.
• 0349 Theorem Theorem 5.2 .
• 034A Proof Proof.
• 034B Subsection 5.2. Dynamical capacity estimates
• 034C Proposition Proposition 5.3 .
• 034D Proof Proof.
• 034E Corollary Corollary 5.4 .
• 034F Proof Proof.
• 034G Corollary Corollary 5.5 .
• 034H Proof Proof.
• 034I Section 6. Appendix: Regularization of qpsh functions
• 034J Theorem Theorem 6.1 .
• 034K Proof Proof.
• 034L Corollary Corollary 6.2 .
• 034M Proof Proof.
• 034N Remark Remark 6.3 .
• 034P Section 1 Introduction
• 034Q Section 2 Superforms and supercurrents on ℝ r
• 034R SourceObject 2.1
• 034S SourceObject 2.2
• 034T SourceObject 2.3
• 034U SourceObject 2.4
• 034V SourceObject 2.5
• 034W SourceObject 2.6
• 034X SourceObject 2.7
• 034Y SourceObject 2.8
• 034Z Proposition Proposition 2.9 (Stokes’ formula)
• 0350 Proof Proof:
• 0351 Proposition Proposition 2.10 (Green’s formula)
• 0352 Proof Proof:
• 0353 SourceObject 2.11
• 0354 Section 3 Superforms on polyhedral complexes
• 0355 SourceObject 3.1
• 0356 SourceObject 3.2
• 0357 SourceObject 3.3
• 0358 SourceObject 3.4
• 0359 Proposition Proposition 3.5 (Stokes’ formula)
• 035A Proof Proof:
• 035B Example Example 3.6
• 035C SourceObject 3.7
• 035D Proposition Proposition 3.8
• 035E Proof Proof:
• 035F SourceObject 3.9
• 035G Proposition Proposition 3.10 (projection formula)
• 035H Proof Proof:
• 035I Section 4 Moment maps and tropical charts
• 035J SourceObject 4.1
• 035K SourceObject 4.2
• 035L Example Example 4.3
• 035M SourceObject 4.4
• 035N Remark Remark 4.5
• 035P SourceObject 4.6
• 035Q SourceObject 4.7
• 035R SourceObject 4.8
• 035S Lemma Lemma 4.9
• 035T Proof Proof:
• 035U SourceObject 4.10
• 035V Proposition Proposition 4.11
• 035W Proof Proof:
• 035X SourceObject 4.12
• 035Y SourceObject 4.13
• 035Z Proposition Proposition 4.14
• 0360 Proof Proof:
• 0361 SourceObject 4.15
• 0362 Proposition Proposition 4.16
• 0363 Proof Proof:
• 0364 Remark Remark 4.17
• 0365 Section 5 Differential forms on algebraic varieties
• 0366 SourceObject 5.1
• 0367 Definition Definition 5.2
• 0368 SourceObject 5.3
• 0369 Remark Remark 5.4
• 036A Definition Definition 5.5
• 036B Proposition Proposition 5.6
• 036C Proof Proof:
• 036D Remark Remark 5.7
• 036E SourceObject 5.8
• 036F Definition Definition 5.9
• 036G Proposition Proposition 5.10
• 036H Proof Proof:
• 036I Lemma Lemma 5.11
• 036J Proof Proof:
• 036K Corollary Corollary 5.12
• 036L Proof Proof:
• 036M Proposition Proposition 5.13
• 036N Proof Proof:
• 036P SourceObject 5.14
• 036Q Lemma Lemma 5.15
• 036R Proof Proof:
• 036S Proposition Proposition 5.16
• 036T Proof Proof:
• 036U Theorem Theorem 5.17
• 036V Proof Proof:
• 036W Remark Remark 5.18
• 036X Section 6 Currents on algebraic varieties
• 036Y SourceObject 6.1
• 036Z SourceObject 6.2
• 0370 Example Example 6.3
• 0371 Remark Remark 6.4
• 0372 Example Example 6.5
• 0373 SourceObject 6.6
• 0374 Example Example 6.7
• 0375 Proposition Proposition 6.8
• 0376 Proof Proof:
• 0377 SourceObject 6.9
• 0378 Proposition Proposition 6.10
• 0379 Proof Proof:
• 037A Section 7 Generalizations to analytic spaces
• 037B SourceObject 7.1
• 037C Proposition Proposition 7.2
• 037D Proof Proof:
• 037E Theorem Theorem 7.3 (Berkovich, Ducros)
• 037F Proof Proof:
• 037G SourceObject 7.4
• 037H Definition Definition 7.5
• 037I Remark Remark 7.6
• 037J Remark Remark 7.7
• 037K Proposition Proposition 7.8
• 037L Proof Proof:
• 037M Proposition Proposition 7.9
• 037N Proof Proof:
• 037P SourceObject 7.10
• 037Q Proposition Proposition 7.11
• 037R Proof Proof:
• 037S Remark Remark 7.12
• 037T Section 1. Introduction
• 037U Theorem Theorem 1.1 .
• 037V Proposition Proposition 1.2 .
• 037W Theorem Theorem 1.3 .
• 037X Theorem Theorem 1.4 .
• 037Y Theorem Theorem 1.5 .
• 037Z Subsection Acknowledgement
• 0380 Section Notations and conventions
• 0381 Section 2. Model metrics, semipositive metrics, and envelopes
• 0382 SourceObject 2.1 .
• 0383 SourceObject 2.2 .
• 0384 SourceObject 2.3 .
• 0385 SourceObject 2.4 .
• 0386 SourceObject 2.5 .
• 0387 SourceObject 2.6 .
• 0388 SourceObject 2.7 .
• 0389 Definition Definition 2.8 .
• 038A Proposition Proposition 2.9 .
• 038B Proof Proof.
• 038C Proposition Proposition 2.10 .
• 038D Proof Proof.
• 038E Proposition Proposition 2.11 .
• 038F Proof Proof.
• 038G Lemma Lemma 2.12 .
• 038H Proof Proof.
• 038I Section 3. Continuity of plurisusbharmonic envelopes on curves
• 038J Theorem Theorem 3.1 .
• 038K SourceObject 3.2 .
• 038L Remark Remark 3.3 .
• 038M Definition Definition 3.4 .
• 038N Proposition Proposition 3.5 .
• 038P Proof Proof.
• 038Q SourceObject 3.6 .
• 038R Proposition Proposition 3.7 .
• 038S Proof Proof.
• 038T Proposition Proposition 3.8 .
• 038U Proof Proof.
• 038V Remark Remark 3.9 .
• 038W Proposition Proposition 3.10 .
• 038X Proof Proof.
• 038Y Proof Proof of Theorem 3.1 .
• 038Z Corollary Corollary 3.11 .
• 0390 Proof Proof.
• 0391 Corollary Corollary 3.12 .
• 0392 Proof Proof.
• 0393 Remark Remark 3.13 .
• 0394 Section 4. Asymptotic test ideals
• 0395 Definition Definition 4.1 .
• 0396 Remark Remark 4.2 .
• 0397 Definition Definition 4.3 .
• 0398 Remark Remark 4.4 .
• 0399 Definition Definition 4.5 .
• 039A Theorem Theorem 4.6 .
• 039B Proof Proof.
• 039C Section 5. Descent for model functions
• 039D Definition Definition 5.1 .
• 039E Proposition Proposition 5.2 .
• 039F Proof Proof.
• 039G Section 6. Resolution of singularities
• 039H Definition Definition 6.1 .
• 039I Definition Definition 6.2 .
• 039J Theorem Theorem 6.3 (Cossart-Piltant) .
• 039K Proof Proof.
• 039L Section 7. Uniform convergence to the envelope of the zero function
• 039M Assumption Assumption 7.1 .
• 039N Remark Remark 7.2 .
• 039P Theorem Theorem 7.3 .
• 039Q Proof Proof.
• 039R Corollary Corollary 7.4 .
• 039S Proof Proof.
• 039T Lemma Lemma 7.5 .
• 039U Proof Proof.
• 039V Section 8. Continuity of the envelope
• 039W Definition Definition 8.1 .
• 039X Theorem Theorem 8.2 .
• 039Y Corollary Corollary 8.3 .
• 039Z Lemma Lemma 8.4 .
• 03A0 Proof Proof.
• 03A1 Lemma Lemma 8.5 .
• 03A2 Proof Proof.
• 03A3 Proof Proof of Theorem 8.2 .
• 03A4 Section 9. The Monge–Ampère equation
• 03A5 Corollary Corollary 9.1 .
• 03A6 Theorem Theorem 9.2 .
• 03A7 Proof Proof.
• 03A8 Theorem Theorem 9.3 .
• 03A9 Proof Proof.
• 03AA Section Appendix A The skeleton and the retraction in the toric case by José Ignacio Burgos Gil and Martín Sombra
• 03AB Question Question 1 .
• 03AC Lemma Lemma A.1 .
• 03AD Proof Proof.
• 03AE Remark Remark A.2 .
• 03AF Theorem Theorem A.3 .
• 03AG Proof Proof.
• 03AH Section 1. Introduction
• 03AI Theorem Theorem 1.1 .
• 03AJ Theorem Theorem 1.2 .
• 03AK Theorem Theorem 1.3 .
• 03AL Corollary Corollary 1.4 .
• 03AM Subsection 1.1. Terminology
• 03AN Subsection 1.2. Acknowledgements
• 03AP Section 2. Formal and piecewise linear metrics
• 03AQ SourceObject 2.1 .
• 03AR SourceObject 2.2 .
• 03AS Remark Remark 2.3 .
• 03AT Definition Definition 2.4 .
• 03AU Remark Remark 2.5 .
• 03AV Proposition Proposition 2.6 .
• 03AW Proof Proof.
• 03AX Definition Definition 2.7 .
• 03AY Proposition Proposition 2.8 .
• 03AZ Proof Proof.
• 03B0 Definition Definition 2.9 .
• 03B1 Proposition Proposition 2.10 .
• 03B2 Proof Proof.
• 03B3 Lemma Lemma 2.11 .
• 03B4 Proof Proof.
• 03B5 Lemma Lemma 2.12 .
• 03B6 Proof Proof.
• 03B7 Proposition Proposition 2.13 .
• 03B8 Proof Proof.
• 03B9 Remark Remark 2.14 .
• 03BA Theorem Theorem 2.15 .
• 03BB Proof Proof.
• 03BC Proposition Proposition 2.16 .
• 03BD Proof Proof.
• 03BE Section 3. Semipositive metrics
• 03BF SourceObject 3.1 .
• 03BG SourceObject 3.2 .
• 03BH Lemma Lemma 3.3 .
• 03BI Proof Proof.
• 03BJ Lemma Lemma 3.4 .
• 03BK Proof Proof.
• 03BL Proposition Proposition 3.5 .
• 03BM Proof Proof.
• 03BN Lemma Lemma 3.6 .
• 03BP Proof Proof.
• 03BQ SourceObject 3.7 .
• 03BR Definition Definition 3.8 .
• 03BS Proposition Proposition 3.9 .
• 03BT Proof Proof.
• 03BU Proposition Proposition 3.10 .
• 03BV Proof Proof.
• 03BW Proposition Proposition 3.11 .
• 03BX Proof Proof.
• 03BY Section 4. Plurisubharmonic model functions
• 03BZ SourceObject 4.1 .
• 03C0 SourceObject 4.2 .
• 03C1 SourceObject 4.3 .
• 03C2 SourceObject 4.4 .
• 03C3 SourceObject 4.5 .
• 03C4 SourceObject 4.6 .
• 03C5 SourceObject 4.7 .
• 03C6 SourceObject 4.8 .
• 03C7 Lemma Lemma 4.9 .
• 03C8 Proof Proof.
• 03C9 Remark Remark 4.10 .
• 03CA Proposition Proposition 4.11 .
• 03CB Proof Proof.
• 03CC Proposition Proposition 4.12 .
• 03CD Proof Proof.
• 03CE Proposition Proposition 4.13 .
• 03CF Proof Proof.
• 03CG Section 5. Semipositivity and pointwise convergence
• 03CH SourceObject 5.1 .
• 03CI Proposition Proposition 5.2 .
• 03CJ Definition Definition 5.3 .
• 03CK Lemma Lemma 5.4 (Step 1 of Lemma 5.12 of [ BFJ16 ] ) .
• 03CL Lemma Lemma 5.5 .
• 03CM Proof Proof.
• 03CN Theorem Theorem 5.6 .
• 03CP Proof Proof.
• 03CQ Corollary Corollary 5.7 .
• 03CR Proof Proof.
• 03CS Section 1. Introduction
• 03CT Acknowledgement Acknowledgments .
• 03CU Section 2. The combinatorial model
• 03CV Subsection 2.1. The base and the discriminant locus
• 03CW Definition Definition .
• 03CX Lemma Lemma 2.1 .
• 03CY Definition Definition .
• 03CZ Definition Definition .
• 03D0 Lemma Lemma 2.2 .
• 03D1 Proof Proof of Lemma 2.2 .
• 03D2 Subsection 2.2. The proof of Lemma 2.1
• 03D3 Definition Definition .
• 03D4 Definition Definition .
• 03D5 Definition Definition .
• 03D6 Lemma Lemma 2.3 .
• 03D7 Proof Proof.
• 03D8 Subsection 2.3. SL ⁡ ( n , ℤ ) -structure and monodromy
• 03D9 Definition Definition .
• 03DA Lemma Lemma 2.4 .
• 03DB Proof Proof.
• 03DC Definition Definition .
• 03DD Theorem Theorem 2.5 .
• 03DE Proof Proof.
• 03DF Subsection 2.4. A glimpse of mirror symmetry
• 03DG Section 3. Torus fibrations of Calabi-Yau toric hypersurfaces
• 03DH Subsection 3.1. The family
• 03DI Remark Remark .
• 03DJ Subsection 3.2. Amoebas of hypersurfaces
• 03DK Definition Definition .
• 03DL Remark Remark .
• 03DM Lemma Lemma 3.1 .
• 03DN Proof Proof.
• 03DP Proposition Proposition 3.2 .
• 03DQ Proof Idea of the proof.
• 03DR Definition Definition .
• 03DS Definition Definition .
• 03DT Lemma Lemma 3.3 .
• 03DU Proof Proof.
• 03DV Subsection 3.3. Foliation of ℝ d \ Q { 0 } λ ​ ( ϵ )
• 03DW Definition Definition .
• 03DX Lemma Lemma 3.4 .
• 03DY Lemma Lemma 3.5 .
• 03DZ Proof Proof.
• 03E0 Remark Remark .
• 03E1 Definition Definition .
• 03E2 Lemma Lemma 3.6 .
• 03E3 Proof Proof.
• 03E4 Remark Remark .
• 03E5 Subsection 3.4. The torus fibration
• 03E6 Definition Definition .
• 03E7 Theorem Theorem 3.7 .
• 03E8 Lemma Lemma 3.8 .
• 03E9 Proof Proof.
• 03EA Remark Remark .
• 03EB Lemma Lemma 3.9 .
• 03EC Proof Proof.
• 03ED Proof Proof of Theorem 3.7 .
• 03EE Remark Remark .
• 03EF Remark Remark .
• 03EG Section 4. Outlook
• 03EH Subsection 4.1. Combinatorics
• 03EI Subsection 4.2. The Hausdorff convergence
• 03EJ Subsection 4.3. Non-Archimedean geometry
• 03EK Section 1. Introduction
• 03EL Notation Notations .
• 03EM Acknowledgement Acknowledgments .
• 03EN Section 2. The model: Kähler affine structures and torus bundles
• 03EP Subsection 2.1. Integral Kähler affine structures
• 03EQ Subsection 2.2. Bi-polyhedral Kähler affine structures
• 03ER Definition Definition .
• 03ES Definition Definition .
• 03ET Definition Definition .
• 03EU Proposition Proposition 2.1 .
• 03EV Proof Proof.
• 03EW Subsection 2.3. The Calabi conjecture
• 03EX Conjecture Conjecture 2.2 (cf. also [ KT02 ] ) .
• 03EY Subsection 2.4. The model torus fibrations as Kähler manifolds
• 03EZ Section 3. Vector fields, foliations and Kähler potentials
• 03F0 Subsection 3.1. Neighborhoods of the discriminant
• 03F1 Subsection 3.2. Regularization
• 03F2 Proposition Proposition 3.1 .
• 03F3 Proof Proof.
• 03F4 Subsection 3.3. Fibration
• 03F5 Lemma Lemma 3.2 .
• 03F6 Proof Proof.
• 03F7 Subsection 3.4. Kähler metrics on the toric variety
• 03F8 Remark Remark .
• 03F9 Lemma Lemma 3.3 .
• 03FA Proof Proof.
• 03FB Proposition Proposition 3.4 .
• 03FC Proof Proof.
• 03FD Section 4. Geometry of Calabi-Yau toric hypersurfaces
• 03FE Subsection 4.1. An embedding of Z a sm into the model torus bundle
• 03FF Lemma Lemma 4.1 .
• 03FG Proof Proof.
• 03FH Lemma Lemma 4.2 .
• 03FI Proof Proof.
• 03FJ Subsection 4.2. Estimates on complex structures and metrics
• 03FK Lemma Lemma 4.3 .
• 03FL Proof Proof.
• 03FM Lemma Lemma 4.4 .
• 03FN Proof Proof.
• 03FP Subsection 4.3. The Gromov-Hausdorff limits of one-parameter families
• 03FQ Theorem Theorem 4.5 .
• 03FR Proof Proof.
• 03FS Remark Remark .
• 03FT Section 1. Introduction
• 03FU Subsection 1.1. Gluing constructions of hyperkähler K3 ⁡ 3 surfaces
• 03FV Subsubsection 1.1.1. Kummer construction
• 03FW Subsubsection 1.1.2. Codimension- 1 collapse
• 03FX Subsubsection 1.1.3. Codimension- 2 collapse
• 03FY Subsubsection 1.1.4. Codimension- 3 collapse with torus fibers
• 03FZ Subsection 1.2. Main results
• 03G0 Theorem Theorem 1.1 .
• 03G1 Remark Remark 1.2 .
• 03G2 Remark Remark 1.3 .
• 03G3 Remark Remark 1.4 .
• 03G4 Theorem Theorem 1.5 .
• 03G5 Remark Remark 1.6 .
• 03G6 Remark Remark 1.7 .
• 03G7 Remark Remark 1.8 .
• 03G8 Remark Remark 1.9 .
• 03G9 Subsection 1.3. Gluing hyperkähler triples
• 03GA Subsection 1.4. Outline of the paper
• 03GB Subsection 1.5. Acknowledgements
• 03GC Section 2. The Gibbons-Hawking ansatz and the model space
• 03GD Example Example 2.1 .
• 03GE Example Example 2.2 .
• 03GF Subsection 2.1. The Heisenberg nilmanifolds
• 03GG Proposition Proposition 2.3 .
• 03GH Proof Proof.
• 03GI Subsection 2.2. The model space
• 03GJ Remark Remark 2.4 .
• 03GK Remark Remark 2.5 .
• 03GL Subsection 2.3. Green’s function on a flat cylinder
• 03GM Theorem Theorem 2.6 .
• 03GN Proof Proof.
• 03GP Corollary Corollary 2.7 .
• 03GQ Remark Remark 2.8 .
• 03GR Section 3. The asymptotic geometry of Tian-Yau spaces
• 03GS Proposition Proposition 3.1 .
• 03GT Proof Proof.
• 03GU Remark Remark 3.2 .
• 03GV Theorem Theorem 3.3 ( [ TY90 , Hei12 ] ) .
• 03GW Proposition Proposition 3.4 .
• 03GX Proof Proof.
• 03GY Remark Remark 3.5 .
• 03GZ Corollary Corollary 3.6 .
• 03H0 Lemma Lemma 3.7 .
• 03H1 Proof Proof.
• 03H2 Section 4. Liouville theorem for harmonic functions
• 03H3 Definition Definition 4.1 .
• 03H4 Example Example 4.2 .
• 03H5 Theorem Theorem 4.3 .
• 03H6 Subsection 4.1. Separation of variables on the model space
• 03H7 Example Example 4.4 (The spectrum of a Heisenberg manifold) .
• 03H8 Subsection 4.2. Uniform estimates for the fundamental solutions
• 03H9 Lemma Lemma 4.5 .
• 03HA Proof Proof.
• 03HB Lemma Lemma 4.6 .
• 03HC Proof Proof.
• 03HD Lemma Lemma 4.7 .
• 03HE Proof Proof.
• 03HF Lemma Lemma 4.8 .
• 03HG Proof Proof.
• 03HH Lemma Lemma 4.9 .
• 03HI Proof Proof.
• 03HJ Subsection 4.3. Asymptotics of harmonic functions
• 03HK Proposition Proposition 4.10 (Harmonic functions with slow exponential growth) .
• 03HL Proof Proof.
• 03HM Lemma Lemma 4.11 .
• 03HN Proof Proof.
• 03HP Subsection 4.4. Regularity and asymptotics for Poisson equation
• 03HQ Lemma Lemma 4.12 .
• 03HR Proof Proof.
• 03HS Lemma Lemma 4.13 .
• 03HT Proof Proof.
• 03HU Lemma Lemma 4.14 (Uniform estimate for eigenfunctions) .
• 03HV Proof Proof.
• 03HW Proposition Proposition 4.15 (Sovability of Poisson Equation) .
• 03HX Proof Proof.
• 03HY Claim Claim 4.16 .
• 03HZ Proof Proof.
• 03I0 Subsection 4.5. Proof of the Liouville theorem
• 03I1 Lemma Lemma 4.17 .
• 03I2 Proof Proof.
• 03I3 Proof Proof of Theorem 4.3 .
• 03I4 Claim Claim 4.18 .
• 03I5 Proof Proof.
• 03I6 Section 5. Liouville theorem for half-harmonic 1-forms
• 03I7 Theorem Theorem 5.1 .
• 03I8 Proof Proof.
• 03I9 Proposition Proposition 5.2 .
• 03IA Remark Remark 5.3 .
• 03IB Proof Proof of Proposition 5.2 .
• 03IC Lemma Lemma 5.4 .
• 03ID Proof Proof.
• 03IE Proposition Proposition 5.5 .
• 03IF Remark Remark 5.6 .
• 03IG Proof Proof of Proposition 5.5 .
• 03IH Section 6. Construction of the approximate hyperkähler triple
• 03II Subsection 6.1. The neck region: a doubly periodic analogue of the Ooguri-Vafa metric
• 03IJ Proposition Proposition 6.1 .
• 03IK Proof Proof.
• 03IL Subsection 6.2. The attaching maps and constraints
• 03IM Remark Remark 6.2 .
• 03IN Remark Remark 6.3 .
• 03IP Subsection 6.3. Gluing definite triples and topology of ℳ
• 03IQ Proposition Proposition 6.4 .
• 03IR Proof Proof.
• 03IS Corollary Corollary 6.5 .
• 03IT Proof Proof.
• 03IU Proposition Proposition 6.6 .
• 03IV Proof Proof.
• 03IW Section 7. Geometry and regularity of the approximate metric
• 03IX Subsection 7.1. Notations
• 03IY Subsubsection 7.1.1. Tian-Yau spaces and their asymptotic rates
• 03IZ Subsubsection 7.1.2. Some notations about the neck region
• 03J0 Subsubsection 7.1.3. Subdivision of the manifold ℳ
• 03J1 Remark Remark 7.1 .
• 03J2 Subsection 7.2. Regularity of the approximate metrics
• 03J3 Lemma Lemma 7.2 .
• 03J4 Remark Remark 7.3 .
• 03J5 Proof Proof.
• 03J6 Theorem Theorem 7.4 (Naber-Zhang, [ NZ16 ] ) .
• 03J7 Remark Remark 7.5 .
• 03J8 Remark Remark 7.6 .
• 03J9 Lemma Lemma 7.7 .
• 03JA Remark Remark 7.8 .
• 03JB Proof Proof.
• 03JC Subsection 7.3. Rescaled geometries
• 03JD Lemma Lemma 7.9 .
• 03JE Remark Remark 7.10 .
• 03JF Proof Proof.
• 03JG Lemma Lemma 7.11 .
• 03JH Proof Proof.
• 03JI Lemma Lemma 7.12 .
• 03JJ Lemma Lemma 7.13 .
• 03JK Remark Remark 7.14 .
• 03JL Proof Proof.
• 03JM Section 8. Weighted Schauder estimate
• 03JN Definition Definition 8.1 (Weighted Hölder space) .
• 03JP Remark Remark 8.2 .
• 03JQ Proposition Proposition 8.3 (The Weighted Schauder Estimate) .
• 03JR Proof Proof.
• 03JS Section 9. Perturbation to genuine hyperkähler metrics
• 03JT Subsection 9.1. The injectivity estimate for 𝒟 g
• 03JU Lemma Lemma 9.1 .
• 03JV Proposition Proposition 9.2 (The Injectivity Estimate for 𝒟 g ) .
• 03JW Proof Proof.
• 03JX Subsection 9.2. The existence of a hyperkähler triple
• 03JY Lemma Lemma 9.3 .
• 03JZ Proposition Proposition 9.4 .
• 03K0 Proof Proof.
• 03K1 Lemma Lemma 9.5 (Nonlinear Errors) .
• 03K2 Proof Proof.
• 03K3 Proposition Proposition 9.6 .
• 03K4 Proof Proof.
• 03K5 Theorem Theorem 9.7 .
• 03K6 Proof Proof.
• 03K7 Subsection 9.3. Completion of main proofs
• 03K8 Proof Proof of Theorem 1.1 .
• 03K9 Proof Proof of Theorem 1.5 .
• 03KA Remark Remark 9.8 .
• 03KB Section 1 Introduction
• 03KC Section 2 Special Lagrangian geometry
• 03KD Subsection 2.1 Special Lagrangian submanifolds in ℂ m
• 03KE Definition Definition 2.1
• 03KF Definition Definition 2.2
• 03KG Proposition Proposition 2.3
• 03KH Proposition Proposition 2.4
• 03KI Proof Proof.
• 03KJ Subsection 2.2 Special Lagrangian m -folds in Calabi–Yau m -folds
• 03KK Definition Definition 2.5
• 03KL Definition Definition 2.6
• 03KM Proposition Proposition 2.7
• 03KN Corollary Corollary 2.8
• 03KP Theorem Theorem 2.9
• 03KQ Theorem Theorem 2.10
• 03KR Subsection 2.3 Almost Calabi–Yau manifolds and genericity
• 03KS Definition Definition 2.11
• 03KT Definition Definition 2.12
• 03KU Theorem Theorem 2.13
• 03KV Section 3 The SYZ Conjecture
• 03KW Subsection 3.1 Why generic SL fibrations cannot be smooth
• 03KX Definition Definition 3.1
• 03KY Proposition Proposition 3.2
• 03KZ Subsection 3.2 Ruan’s Lagrangian fibrations by gradient flow
• 03L0 Theorem Theorem 3.3
• 03L1 Section 4 Two simple SL fibrations of ℂ 3
• 03L2 Theorem Theorem 4.1
• 03L3 Corollary Corollary 4.2
• 03L4 Theorem Theorem 4.3
• 03L5 Corollary Corollary 4.4
• 03L6 Section 5 Two piecewise smooth SL fibrations of ℂ 3
• 03L7 Definition Definition 5.1
• 03L8 Theorem Theorem 5.2
• 03L9 Proof Proof.
• 03LA Definition Definition 5.3
• 03LB Theorem Theorem 5.4
• 03LC Example Example 5.5
• 03LD Section 6 A class of U ( 1 ) -invariant SL 3-folds in ℂ 3
• 03LE Subsection 6.1 Finding the equations on u and v
• 03LF Proposition Proposition 6.1
• 03LG Proof Proof.
• 03LH Lemma Lemma 6.2
• 03LI Proposition Proposition 6.3
• 03LJ Proposition Proposition 6.4
• 03LK Proposition Proposition 6.5
• 03LL Proof Proof.
• 03LM Subsection 6.2 Writing the fibrations of § 4 and § 5 in this form
• 03LN Lemma Lemma 6.6
• 03LP Proposition Proposition 6.7
• 03LQ Proof Proof.
• 03LR Proposition Proposition 6.8
• 03LS Subsection 6.3 Other examples of solutions to ( 32 ) and ( 33 )
• 03LT Example Example 6.9
• 03LU Example Example 6.10
• 03LV Example Example 6.11
• 03LW Subsection 6.4 Isolated singularities of solutions to ( 32 )
• 03LX Definition Definition 6.12
• 03LY Lemma Lemma 6.13
• 03LZ Conjecture Conjecture 6.14
• 03M0 Subsection 6.5 Geometric interpretation
• 03M1 Section 7 Higher-order singularities of SL fibrations
• 03M2 Subsection 7.1 A conjectural local model for SL fibrations
• 03M3 Assumption Assumption 7.1
• 03M4 Assumption Assumption 7.2
• 03M5 Assumption Assumption 7.3
• 03M6 Conjecture Conjecture 7.4
• 03M7 Subsection 7.2 Justification for the conjecture
• 03M8 Lemma Lemma 7.5
• 03M9 Proof Proof.
• 03MA Subsection 7.3 Holomorphic discs with boundary in N a , b , c
• 03MB Proposition Proposition 7.6
• 03MC Proof Proof.
• 03MD Section 8 A model of a ‘ribbon’ in the discriminant
• 03ME Subsection 8.1 A variation on the fibration of § 7.1
• 03MF Assumption Assumption 8.1
• 03MG Assumption Assumption 8.2
• 03MH Assumption Assumption 8.3
• 03MI Subsection 8.2 Monodromy around a ‘ribbon’
• 03MJ Section 9 Global behaviour of SL fibrations
• 03MK Subsection 9.1 The Gross–Ruan picture of smooth SL fibrations
• 03ML Subsection 9.2 Modification of this picture for generic ACY 3-folds
• 03MM Subsection 9.3 Conclusions
• 03MN Conjecture Conjecture 9.1
• 03MP Section 1 Introduction
• 03MQ Section 2 Background material
• 03MR Subsection 2.1 Calabi–Yau m -folds and special Lagrangians
• 03MS Definition Definition 2.1 .
• 03MT Theorem Theorem 2.2 .
• 03MU Subsection 2.2 Special Lagrangian m -folds in ℂ m
• 03MV Definition Definition 2.3 .
• 03MW Theorem Theorem 2.4 .
• 03MX Example Example 2.5 .
• 03MY Theorem Theorem 2.6 .
• 03MZ Example Example 2.7 .
• 03N0 Example Example 2.8 .
• 03N1 Subsection 2.3 Lagrangian mean curvature flow
• 03N2 Definition Definition 2.9 .
• 03N3 Definition Definition 2.10 .
• 03N4 Theorem Theorem 2.11 .
• 03N5 Theorem Theorem 2.12 .
• 03N6 Subsection 2.4 Examples of solitons for Lagrangian MCF
• 03N7 Example Example 2.13 .
• 03N8 Theorem Theorem 2.14 .
• 03N9 Example Example 2.15 .
• 03NA Example Example 2.16 .
• 03NB Subsection 2.5 Lagrangian Floer cohomology and Fukaya categories
• 03NC Definition Definition 2.17 .
• 03ND Definition Definition 2.18 .
• 03NE Remark Remark 2.19 .
• 03NF Definition Definition 2.20 .
• 03NG Lemma Lemma 2.21 .
• 03NH Remark Remark 2.22 .
• 03NI Subsection 2.6 H ​ F ∗ and D b ℱ ( M ) for immersed Lagrangians
• 03NJ Lemma Lemma 2.23 .
• 03NK Remark Remark 2.24 .
• 03NL Section 3 The conjectures
• 03NM Subsection 3.1 Bridgeland stability on D b ℱ ( M ) for M Calabi–Yau
• 03NN Definition Definition 3.1 .
• 03NP Conjecture Conjecture 3.2 .
• 03NQ Remark Remark 3.3 .
• 03NR Definition Definition 3.4 .
• 03NS Conjecture Conjecture 3.5 .
• 03NT Conjecture Conjecture 3.6 .
• 03NU Remark Remark 3.7 .
• 03NV Subsection 3.2 Approaching Conjecture 3.2 using Lagrangian MCF
• 03NW Remark Remark 3.8 .
• 03NX Subsection 3.3 On finite time singularities of Lagrangian MCF
• 03NY SourceObject Principle 3.9 .
• 03NZ Remark Remark 3.10 .
• 03P0 SourceObject Principle 3.11 .
• 03P1 SourceObject Problem 3.12 .
• 03P2 Subsection 3.4 Flowing from unobstructed to obstructed immersed Lagrangians
• 03P3 Remark Remark 3.13 .
• 03P4 SourceObject Problem 3.14 .
• 03P5 Example Example 3.15 .
• 03P6 Subsection 3.5 ‘Neck pinches’ using Lawlor necks
• 03P7 Conjecture Conjecture 3.16 .
• 03P8 Remark Remark 3.17 .
• 03P9 Subsection 3.6 Including singular Lagrangians in D b ℱ ( M ) ; LMCF for Lagrangians with stable conical singularities
• 03PA SourceObject Principle 3.18 .
• 03PB Theorem Theorem 3.19 .
• 03PC SourceObject Problem 3.20 .
• 03PD Remark Remark 3.21 .
• 03PE Subsection 3.7 Collapsing zero objects in D b ℱ ( M )
• 03PF Example Example 3.22 .
• 03PG Example Example 3.23 .
• 03PH Example Example 3.24 .
• 03PI SourceObject Principle 3.25 .
• 03PJ Lemma Lemma 3.26 .
• 03PK SourceObject Problem 3.27 .
• 03PL Example Example 3.28 .
• 03PM Subsection 3.8 What goes wrong in LMCF of obstructed Lagrangians
• 03PN SourceObject Principle 3.29 .
• 03PP Remark Remark 3.30 .
• 03PQ Example Example 3.31 .
• 03PR Example Example 3.32 .
• 03PS Conjecture Conjecture 3.33 .
• 03PT Subsection 3.9 A Thomas–Yau type conjecture
• 03PU Conjecture Conjecture 3.34 .
• 03PV Remark Remark 3.35 .
• 03PW Theorem Theorem 1.1
• 03PX Theorem Theorem 2.1
• 03PY Proof Proof
• 03PZ Proposition Proposition
• 03Q0 Proof Proof
• 03Q1 Section 1 Introduction
• 03Q2 Subsection 1.1 Homological mirror symmetry and degenerations
• 03Q3 Subsection 1.2 Content of the paper
• 03Q4 Section 2 Degenerations of unitary Conformal Field Theories
• 03Q5 Remark Remark 1
• 03Q6 Subsection 2.1 Moduli space of Conformal Field Theories
• 03Q7 Remark Remark 2
• 03Q8 Subsection 2.2 Physical picture of a simple collapse
• 03Q9 Remark Remark 3
• 03QA Subsection 2.3 Multiple collapse and the structure of the boundary
• 03QB Subsection 2.4 Example: Toroidal models
• 03QC Subsection 2.5 Example: WZW model for S ​ U ​ ( 2 )
• 03QD Subsection 2.6 A-model and B-model of N = 2 SCFT as boundary strata
• 03QE Subsection 2.7 Mirror symmetry and the collapse
• 03QF Section 3 Calabi-Yau manifolds in the large complex structure limit
• 03QG Subsection 3.1 Maximal degenerations of Calabi-Yau manifolds
• 03QH Definition Definition 1
• 03QI Lemma Lemma 1
• 03QJ Conjecture Conjecture 1
• 03QK Remark Remark 4
• 03QL Conjecture Conjecture 2
• 03QM Remark Remark 5
• 03QN Subsection 3.2 Monge-Ampère manifolds and duality of torus fibrations
• 03QP Definition Definition 2
• 03QQ Lemma Lemma 2
• 03QR Proof Proof.
• 03QS Proposition Proposition 1
• 03QT Corollary Corollary 1
• 03QU Conjecture Conjecture 3
• 03QV Remark Remark 6
• 03QW Remark Remark 7
• 03QX Subsection 3.3 Speculations about relations with non-archimedean geometry
• 03QY Section 4 A ∞ -algebras and A ∞ -categories
• 03QZ Subsection 4.1 Two problems with the general definition
• 03R0 Subsection 4.2 Non-unital A ∞ -algebras and A ∞ -categories
• 03R1 Definition Definition 3
• 03R2 Definition Definition 4
• 03R3 Definition Definition 5
• 03R4 Remark Remark 8
• 03R5 Definition Definition 6
• 03R6 Remark Remark 9
• 03R7 Subsection 4.3 A ∞ -pre-categories
• 03R8 Definition Definition 7
• 03R9 Definition Definition 8
• 03RA Definition Definition 9
• 03RB Definition Definition 10
• 03RC Remark Remark 10
• 03RD Definition Definition 11
• 03RE Definition Definition 12
• 03RF Definition Definition 13
• 03RG Definition Definition 14
• 03RH Conjecture Conjecture 4
• 03RI Subsection 4.4 Example: directed A ∞ -pre-categories
• 03RJ Definition Definition 15
• 03RK Conjecture Conjecture 5
• 03RL Section 5 Fukaya category and its degeneration
• 03RM Subsection 5.1 Fukaya category
• 03RN Definition Definition 16
• 03RP Definition Definition 17
• 03RQ Remark Remark 11
• 03RR Proposition Proposition 2
• 03RS Theorem Theorem 1
• 03RT Subsection 5.2 Fukaya-Oh category for torus fibration
• 03RU Remark Remark 12
• 03RV Proposition Proposition 3
• 03RW Remark Remark 13
• 03RX Remark Remark 14
• 03RY Section 6 Morse-Smale complex and the category of Morse functions
• 03RZ Subsection 6.1 Notations from Morse theory
• 03S0 Subsection 6.2 Morse A ∞ -category of smooth functions
• 03S1 Definition Definition 18
• 03S2 Remark Remark 15
• 03S3 Definition Definition 19
• 03S4 Definition Definition 20
• 03S5 Subsection 6.3 De Rham A ∞ -category of smooth functions
• 03S6 Theorem Theorem 2
• 03S7 Subsection 6.4 A ∞ -structure on a subcomplex
• 03S8 Proposition Proposition 4
• 03S9 Theorem Theorem 3
• 03SA Proposition Proposition 5
• 03SB Remark Remark 16
• 03SC Subsection 6.5 Projectors and homotopies in Morse theory
• 03SD Lemma Lemma 3
• 03SE Proof Proof .
• 03SF Proposition Proposition 6
• 03SG Remark Remark 17
• 03SH Subsection 6.6 Proof of the theorem
• 03SI Section 7 A ∞ -structure for the derived category of coherent sheaves
• 03SJ Subsection 7.1 Rigid analytic space
• 03SK Definition Definition 21
• 03SL Definition Definition 22
• 03SM Proposition Proposition 7
• 03SN Remark Remark 18
• 03SP Subsection 7.2 A ∞ -structure on the derived category of coherent sheaves
• 03SQ Proposition Proposition 8
• 03SR Section 8 Homological mirror conjecture
• 03SS Subsection 8.1 Mirror symmetry functor on objects
• 03ST Subsection 8.2 Spectrum of a morphism and the semigroup
• 03SU Remark Remark 19
• 03SV Proposition Proposition 9
• 03SW Proposition Proposition 10
• 03SX Subsection 8.3 Homological mirror symmetry for abelian varieties
• 03SY Proposition Proposition 11
• 03SZ Proposition Proposition 12
• 03T0 Theorem Theorem 4
• 03T1 Remark Remark 20
• 03T2 Section 9 Appendix: constructions in the case of complex numbers
• 03T3 Subsection 9.1 Mirror symmetry functor on objects over 𝐂
• 03T4 Lemma Lemma 4
• 03T5 Proof Proof.
• 03T6 Proposition Proposition 13
• 03T7 Proof Proof.
• 03T8 Definition Definition 23
• 03T9 Subsection 9.2 Sectors in the space of Dolbeault forms
• 03TA Remark Remark 21
• 03TB Subsection 9.3 Semigroup φ t
• 03TC Section 1 Introduction
• 03TD Subsection 1.1
• 03TE Subsection 1.2
• 03TF Subsection 1.3
• 03TG Subsection 1.4
• 03TH Section Part I
• 03TI Section 2 𝐙 -affine structures
• 03TJ Subsection 2.1 Definitions
• 03TK Definition Definition 1
• 03TL Definition Definition 2
• 03TM Definition Definition 3
• 03TN Subsection 2.2 Monodromy representation and its invariant
• 03TP Section 3 A-model construction
• 03TQ Subsection 3.1 Integrable systems
• 03TR Subsubsection 3.1.1 Cohomological interpretation of class [ ρ ]
• 03TS Subsection 3.2 Examples of integrable systems
• 03TT Subsubsection 3.2.1 Flat tori
• 03TU Subsubsection 3.2.2 Surfaces
• 03TV Subsubsection 3.2.3 Moment map
• 03TW Subsubsection 3.2.4 K3 surfaces
• 03TX Subsection 3.3 Families of integrable systems and PL actions
• 03TY Section 4 B-model construction
• 03TZ Subsection 4.1 𝐙 -affine structure on smooth points
• 03U0 Definition Definition 4
• 03U1 Theorem Theorem 1
• 03U2 Lemma Lemma 1
• 03U3 Subsection 4.2 Examples
• 03U4 Subsubsection 4.2.1 Logarithmic map
• 03U5 Subsubsection 4.2.2 Tate tori
• 03U6 Subsubsection 4.2.3 Clemens polytopes and their contractions
• 03U7 Subsubsection 4.2.4 Curves
• 03U8 Subsubsection 4.2.5 K3 surfaces
• 03U9 Subsection 4.3 Stein property
• 03UA Proposition Proposition 1
• 03UB Proof Proof.
• 03UC Section 5 𝐙 -affine structures and mirror symmetry
• 03UD Subsection 5.1 Gromov-Hausdorff collapse of Calabi-Yau manifolds
• 03UE Definition Definition 5
• 03UF Conjecture Conjecture 1
• 03UG Definition Definition 6
• 03UH Proposition Proposition 2
• 03UI Corollary Corollary 1
• 03UJ Conjecture Conjecture 2
• 03UK Subsubsection 5.1.1 K3 example
• 03UL Subsection 5.2 Non-archimedean picture for the space B
• 03UM Conjecture Conjecture 3
• 03UN Conjecture Conjecture 4
• 03UP Section Part II
• 03UQ Section 6 Compactifications of 𝐙 -affine structures
• 03UR Subsection 6.1 Properties of compactifications
• 03US Subsection 6.2 PL compactifications
• 03UT Proposition Proposition 3
• 03UU Proof Proof.
• 03UV Subsection 6.3 Some conjectures about singular sets
• 03UW Conjecture Conjecture 5
• 03UX Definition Definition 7
• 03UY Definition Definition 8
• 03UZ Conjecture Conjecture 6
• 03V0 Subsection 6.4 Standard singularities in codimension two
• 03V1 Remark Remark 1
• 03V2 Subsection 6.5 𝐙 -affine version of Gauss-Bonnet theorem
• 03V3 Theorem Theorem 2
• 03V4 Proof Proof of the Theorem
• 03V5 Corollary Corollary 2
• 03V6 Proof Proof.
• 03V7 Remark Remark 2
• 03V8 Subsection 6.6 Skeleton of a non-archimedean Calabi-Yau variety
• 03V9 Definition Definition 9
• 03VA Definition Definition 10
• 03VB Theorem Theorem 3
• 03VC Subsection 6.7 K3 surfaces and 𝐙 ​ P ​ L -actions on S 2
• 03VD Subsubsection 6.7.1 Integrable systems
• 03VE Conjecture Conjecture 7
• 03VF Subsubsection 6.7.2 Analytic surfaces
• 03VG Conjecture Conjecture 8
• 03VH Conjecture Conjecture 9
• 03VI Subsubsection 6.7.3 Lattice points
• 03VJ Subsection 6.8 Further examples
• 03VK Section 7 K -affine structures
• 03VL Subsection 7.1 Definitions
• 03VM Definition Definition 11
• 03VN Definition Definition 12
• 03VP Subsection 7.2 K -affine structure on smooth points
• 03VQ Lemma Lemma 2
• 03VR Proof Proof:
• 03VS Theorem Theorem 4
• 03VT Proof Proof.
• 03VU Subsection 7.3 Lifting Problem
• 03VV Remark Remark 3
• 03VW Subsection 7.4 Flat coordinates and periods
• 03VX Subsubsection 7.4.1 Flat coordinates for degenerating complex Calabi-Yau manifolds
• 03VY Subsubsection 7.4.2 Non-archimedean periods
• 03VZ Conjecture Conjecture 10
• 03W0 Section Part III
• 03W1 Theorem Theorem 5
• 03W2 Section 8 Model near a singular point
• 03W3 Proposition Proposition 4
• 03W4 Proof Proof.
• 03W5 Lemma Lemma 3
• 03W6 Proof Proof:
• 03W7 Section 9 Lines on surfaces
• 03W8 Subsection 9.1 Data
• 03W9 Subsection 9.2 Axioms
• 03WA Subsection 9.3 Example: gradient lines
• 03WB Section 10 Groups and symplectomorphisms
• 03WC Subsection 10.1 Pro-nilpotent Lie algebra
• 03WD Subsection 10.2 Lie groups G λ
• 03WE Lemma Lemma 4
• 03WF Proof Proof.
• 03WG Theorem Theorem 6
• 03WH Proof Proof.
• 03WI Subsection 10.3 Function o ​ r ​ d l
• 03WJ Subsection 10.4 Symplectomorphisms assigned to lines
• 03WK Section 11 Modification of the sheaf 𝒪 c ​ a ​ n
• 03WL Subsection 11.1 Pieces of lines and convergence regions
• 03WM Definition Definition 13
• 03WN Definition Definition 14
• 03WP Proposition Proposition 5
• 03WQ Subsection 11.2 Main assumptions, and an apology
• 03WR Subsection 11.3 Infinite product and its convergence
• 03WS Theorem Theorem 7
• 03WT Proof Proof.
• 03WU Theorem Theorem 8
• 03WV Lemma Lemma 5
• 03WW Subsection 11.4 Construction of the modified sheaf 𝒪 B m ​ o ​ d ​ i ​ f
• 03WX Proposition Proposition 6
• 03WY Proof Proof.
• 03WZ Subsection 11.5 Construction of the collection of lines
• 03X0 Proposition Proposition 7
• 03X1 Lemma Lemma 6
• 03X2 Proof Proof.
• 03X3 Lemma Lemma 7
• 03X4 Proof Proof:
• 03X5 Subsection 11.6 Independence and uniqueness
• 03X6 Conjecture Conjecture 11
• 03X7 Remark Remark 4
• 03X8 Subsection 11.7 Remark on the case of positive and mixed characteristic
• 03X9 Subsection 11.8 Further generalizations
• 03XA Section Appendix A Analytic geometry
• 03XB Subsection A.1 Berkovich spectrum
• 03XC Definition Definition 15
• 03XD Definition Definition 16
• 03XE Definition Definition 17
• 03XF Proposition Proposition 8
• 03XG Example Example 1
• 03XH Subsection A.2 Algebraic torus and the logarithmic map
• 03XI Subsection A.3 Clemens polytopes
• 03XJ Definition Definition 18
• 03XK Definition Definition 19
• 03XL Definition Definition 20
• 03XM Definition Definition 21
• 03XN Subsection A.4 Simple blow-ups
• 03XP Definition Definition 22
• 03XQ Theorem Theorem 9
• 03XR Corollary Corollary 3
• 03XS Subsection A.5 Clemens cones and valuations
• 03XT Proposition Proposition 9
• 03XU Subsection A.6 Clemens cones and paths
• 03XV Proposition Proposition 10
• 03XW Lemma Lemma 8
• 03XX Corollary Corollary 4
• 03XY Theorem Theorem 10
• 03XZ Section Appendix B Torelli theorem for K3 surfaces
• 03Y0 Definition Definition 23
• 03Y1 Theorem Theorem 11
• 03Y2 Section Chapter 1 Introduction and Background Review
• 03Y3 Conjecture Conjecture 1.1 .
• 03Y4 Theorem Theorem 1.2 .
• 03Y5 Remark Remark 1.1 .
• 03Y6 Theorem Theorem 1.3 .
• 03Y7 Theorem Theorem 1.4 .
• 03Y8 Remark Remark 1.2 .
• 03Y9 Remark Remark 1.3 .
• 03YA Notation Notation .
• 03YB Section 1.1. Gross-Ruan-Joyce picture of SYZ fibrations
• 03YC Subsection 1.1.1. Generic region
• 03YD Subsection 1.1.2. Edges
• 03YE Remark Remark 1.4 .
• 03YF Subsection 1.1.3. Positive vertices
• 03YG Subsection 1.1.4. Negative vertices
• 03YH Subsection 1.1.5. Joyce’s critique
• 03YI Subsection 1.1.6. Degenerating toric Calabi-Yau hypersurfaces
• 03YJ Example Example 1.1 .
• 03YK Example Example 1.2 .
• 03YL Example Example 1.3 .
• 03YM Example Example 1.4 .
• 03YN Example Example 1.5 .
• 03YP Section 1.2. Generalised Gibbons-Hawking ansatz
• 03YQ Theorem Theorem 1.5 .
• 03YR Remark Remark 1.5 .
• 03YS Proof Proof.
• 03YT Remark Remark 1.6 .
• 03YU Remark Remark 1.7 .
• 03YV Remark Remark 1.8 .
• 03YW Subsection 1.2.1. Elementary examples
• 03YX Example Example 1.6 .
• 03YY Example Example 1.7 .
• 03YZ Lemma Lemma 1.6 .
• 03Z0 Proof Proof.
• 03Z1 Remark Remark 1.9 .
• 03Z2 Example Example 1.8 .
• 03Z3 Subsection 1.2.2. Compactification and distributional equation
• 03Z4 Example Example 1.9 .
• 03Z5 Example Example 1.10 .
• 03Z6 Section 1.3. Ooguri-Vafa metric
• 03Z7 Subsection 1.3.1. Gibbons-Hawking viewpoint
• 03Z8 Subsection 1.3.2. Holomorphic viewpoint
• 03Z9 Remark Remark 1.10 .
• 03ZA Subsection 1.3.3. Some conceptual aspects of the Ooguri-Vafa metrics
• 03ZB Section 1.4. Synopsis of the new metrics
• 03ZC Subsection 1.4.1. Geometric aspects
• 03ZD Subsection 1.4.2. Analytic aspects
• 03ZE Subsection 1.4.3. Outlook: towards the SYZ conjecture
• 03ZF Section Chapter 2 Taub-NUT Type Metrics on ℂ 3
• 03ZG Section 2.1. First order asymptotic metric near infinity
• 03ZH Remark Remark 2.1 .
• 03ZI Lemma Lemma 2.1 .
• 03ZJ Proof Proof.
• 03ZK Proposition Proposition 2.2 .
• 03ZL Proof Proof.
• 03ZM Remark Remark 2.2 .
• 03ZN Remark Remark 2.3 .
• 03ZP Section 2.2. Metric behaviour away from the discriminant locus
• 03ZQ Lemma Lemma 2.3 .
• 03ZR Proof Proof.
• 03ZS Corollary Corollary 2.4 .
• 03ZT Section 2.3. Structure near discriminant locus
• 03ZU Remark Remark 2.4 .
• 03ZV Lemma Lemma 2.5 .
• 03ZW Proof Proof.
• 03ZX Lemma Lemma 2.6 .
• 03ZY Remark Remark 2.5 .
• 03ZZ Remark Remark 2.6 .
• 0400 Remark Remark 2.7 .
• 0401 Section 2.4. Complex geometric perspective
• 0402 Lemma Lemma 2.7 .
• 0403 Proof Proof.
• 0404 Lemma Lemma 2.8 .
• 0405 Proof Proof.
• 0406 Lemma Lemma 2.9 .
• 0407 Proof Proof.
• 0408 Lemma Lemma 2.10 .
• 0409 Proof Proof.
• 040A Proposition Proposition 2.11 .
• 040B Proof Proof.
• 040C Remark Remark 2.8 .
• 040D Section 2.5. Algebraic geometric perspective
• 040E Lemma Lemma 2.12 .
• 040F Proof Proof.
• 040G Proposition Proposition 2.13 .
• 040H Proof Proof.
• 040I Section 2.6. Surgery on the ansatz
• 040J Lemma Lemma 2.14 .
• 040K Section 2.7. Hein’s package and weighted Sobolev inequality
• 040L Proposition Proposition 2.15 .
• 040M Proof Proof.
• 040N Corollary Corollary 2.16 .
• 040P Corollary Corollary 2.17 .
• 040Q Section 2.8. Harmonic analysis
• 040R Lemma Lemma 2.18 .
• 040S Proof Proof.
• 040T Lemma Lemma 2.19 .
• 040U Proof Proof.
• 040V Lemma Lemma 2.20 .
• 040W Proof Proof.
• 040X Lemma Lemma 2.21 .
• 040Y Proof Proof.
• 040Z Lemma Lemma 2.22 .
• 0410 Proof Proof.
• 0411 Proposition Proposition 2.23 .
• 0412 Proof Proof.
• 0413 Remark Remark 2.9 .
• 0414 Remark Remark 2.10 .
• 0415 Corollary Corollary 2.24 .
• 0416 Proof Proof.
• 0417 Section 2.9. Perturbation into a Calabi-Yau metric
• 0418 Lemma Lemma 2.25 .
• 0419 Proof Proof.
• 041A Theorem Theorem 2.26 .
• 041B Proof Proof.
• 041C Corollary Corollary 2.27 .
• 041D Corollary Corollary 2.28 .
• 041E Proof Proof.
• 041F Corollary Corollary 2.29 .
• 041G Remark Remark 2.11 .
• 041H Proof Proof.
• 041I Section 2.10. Uniqueness and moduli
• 041J Lemma Lemma 2.30 .
• 041K Proof Proof.
• 041L Proposition Proposition 2.31 .
• 041M Proof Proof.
• 041N Question Question .
• 041P Question Question .
• 041Q Section 2.11. Exotic metrics: past and future
• 041R Subsection 2.11.1. Exotic metrics on ℂ n
• 041S Question Question .
• 041T Subsection 2.11.2. Gravitational instantons
• 041U Remark Remark 2.12 .
• 041V Subsection 2.11.3. Generalisation of ALF geometry
• 041W Subsection 2.11.4. Connection to collapsing compact Calabi-Yau metrics
• 041X Section Chapter 3 The Positive Vertex
• 041Y Section 3.1. First order approximate metric
• 041Z Notation Notation .
• 0420 Proposition Proposition 3.1 .
• 0421 Proof Proof.
• 0422 Remark Remark 3.1 .
• 0423 Remark Remark 3.2 .
• 0424 Remark Remark 3.3 .
• 0425 Lemma Lemma 3.2 .
• 0426 Proof Proof.
• 0427 Section 3.2. Asymptotes for the first order ansatz
• 0428 Proposition Proposition 3.3 .
• 0429 Proof Proof.
• 042A Lemma Lemma 3.4 .
• 042B Proof Proof.
• 042C Proposition Proposition 3.5 .
• 042D Proof Proof.
• 042E Remark Remark 3.4 .
• 042F Lemma Lemma 3.6 .
• 042G Proof Proof.
• 042H Section 3.3. Complex geometric perspective
• 042I Lemma Lemma 3.7 .
• 042J Proof Proof.
• 042K Lemma Lemma 3.8 .
• 042L Proof Proof.
• 042M Lemma Lemma 3.9 .
• 042N Proof Proof.
• 042P Lemma Lemma 3.10 .
• 042Q Proof Proof.
• 042R Proposition Proposition 3.11 .
• 042S Proof Proof.
• 042T Remark Remark 3.5 .
• 042U Section 3.4. Weighted Hölder norms and initial error estimates
• 042V Notation Notation .
• 042W Notation Notation .
• 042X Lemma Lemma 3.12 .
• 042Y Lemma Lemma 3.13 .
• 042Z Lemma Lemma 3.14 .
• 0430 Lemma Lemma 3.15 .
• 0431 Lemma Lemma 3.16 .
• 0432 Section 3.5. Harmonic analysis I: periodic Euclidean region
• 0433 Lemma Lemma 3.17 .
• 0434 Proof Proof.
• 0435 Lemma Lemma 3.18 .
• 0436 Proof Proof.
• 0437 Lemma Lemma 3.19 .
• 0438 Remark Remark 3.6 .
• 0439 Remark Remark 3.7 .
• 043A Proof Proof.
• 043B Lemma Lemma 3.20 .
• 043C Proof Proof.
• 043D Proposition Proposition 3.21 .
• 043E Proposition Proposition 3.22 .
• 043F Remark Remark 3.8 .
• 043G Section 3.6. Perturbation in the Euclidean region
• 043H Proposition Proposition 3.23 .
• 043I Proof Proof.
• 043J Corollary Corollary 3.24 .
• 043K Section 3.7. Glue in the Taub-NUT type metric on ℂ 3
• 043L Subsection 3.7.1. Relative Gibbons-Hawking potential
• 043M Lemma Lemma 3.25 .
• 043N Proof Proof.
• 043P Lemma Lemma 3.26 .
• 043Q Proof Proof.
• 043R Corollary Corollary 3.27 .
• 043S Proof Proof.
• 043T Subsection 3.7.2. Modifying the Kähler ansatz I
• 043U Lemma Lemma 3.28 .
• 043V Proposition Proposition 3.29 .
• 043W Subsection 3.7.3. Modifying the Kähler ansatz II
• 043X Lemma Lemma 3.30 .
• 043Y Subsection 3.7.4. Global weighted Hölder norms and error estimates
• 043Z Proposition Proposition 3.31 .
• 0440 Proof Proof.
• 0441 Section 3.8. Harmonic analysis II: perturbation to Calabi-Yau metric
• 0442 Proposition Proposition 3.32 .
• 0443 Proof Proof.
• 0444 Theorem Theorem 3.33 .
• 0445 Proof Proof.
• 0446 Remark Remark 3.9 .
• 0447 Section 3.9. Ooguri-Vafa type metric on the positive vertex
• 0448 Corollary Corollary 3.34 .
• 0449 Corollary Corollary 3.35 .
• 044A Proof Proof.
• 044B Remark Remark 3.10 .
• 044C Section 3.10. Incompleteness and running coupling
• 044D Proposition Proposition 3.36 .
• 044E Proof Proof.
• 044F Section Chapter 4 The Negative Vertex
• 044G Section 4.1. First order approximate metric
• 044H Notation Notation .
• 044I Remark Remark 4.1 .
• 044J Lemma Lemma 4.1 .
• 044K Proof Proof.
• 044L Lemma Lemma 4.2 .
• 044M Proof Proof.
• 044N Lemma Lemma 4.3 .
• 044P Proof Proof.
• 044Q Proposition Proposition 4.4 .
• 044R Lemma Lemma 4.5 .
• 044S Proof Proof.
• 044T Corollary Corollary 4.6 .
• 044U Proof Proof.
• 044V Proposition Proposition 4.7 .
• 044W Remark Remark 4.2 .
• 044X Section 4.2. Asymptotic for the first order ansatz I
• 044Y Lemma Lemma 4.8 .
• 044Z Proof Proof.
• 0450 Corollary Corollary 4.9 .
• 0451 Lemma Lemma 4.10 .
• 0452 Proof Proof.
• 0453 Lemma Lemma 4.11 .
• 0454 Remark Remark 4.3 .
• 0455 Section 4.3. Asymptotic for the first order ansatz II
• 0456 Lemma Lemma 4.12 .
• 0457 Proof Proof.
• 0458 Proposition Proposition 4.13 .
• 0459 Proof Proof.
• 045A Lemma Lemma 4.14 .
• 045B Proof Proof.
• 045C Section 4.4. Structure near the singular locus I
• 045D Lemma Lemma 4.15 .
• 045E Proof Proof.
• 045F Proposition Proposition 4.16 .
• 045G Remark Remark 4.4 .
• 045H Proof Proof.
• 045I Corollary Corollary 4.17 .
• 045J Section 4.5. Structure near the singular locus II
• 045K Proposition Proposition 4.18 .
• 045L Proof Proof.
• 045M Proposition Proposition 4.19 .
• 045N Proof Proof.
• 045P Remark Remark 4.5 .
• 045Q Section 4.6. Complex geometric perspective
• 045R Lemma Lemma 4.20 .
• 045S Proof Proof.
• 045T Lemma Lemma 4.21 .
• 045U Proof Proof.
• 045V Lemma Lemma 4.22 .
• 045W Proof Proof.
• 045X Lemma Lemma 4.23 .
• 045Y Proof Proof.
• 045Z Lemma Lemma 4.24 .
• 0460 Proof Proof.
• 0461 Lemma Lemma 4.25 .
• 0462 Proof Proof.
• 0463 Remark Remark 4.6 .
• 0464 Remark Remark 4.7 .
• 0465 Proposition Proposition 4.26 .
• 0466 Proof Proof.
• 0467 Lemma Lemma 4.27 .
• 0468 Proof Proof.
• 0469 Lemma Lemma 4.28 .
• 046A Proof Proof.
• 046B Proposition Proposition 4.29 .
• 046C Proof Proof.
• 046D Section 4.7. Weighted Hölder norms and initial error estimate
• 046E Notation Notation .
• 046F Notation Notation .
• 046G Lemma Lemma 4.30 .
• 046H Proof Proof.
• 046I Section 4.8. Harmonic analysis I: periodic Euclidean region
• 046J Proposition Proposition 4.31 .
• 046K Remark Remark 4.8 .
• 046L Proposition Proposition 4.32 .
• 046M Proposition Proposition 4.33 .
• 046N Proposition Proposition 4.34 .
• 046P Proof Proof.
• 046Q Section 4.9. Harmonic analysis II: Neighbourhood of S
• 046R Lemma Lemma 4.35 .
• 046S Proof Proof.
• 046T Lemma Lemma 4.36 .
• 046U Proof Proof.
• 046V Section 4.10. Harmonic analysis III: perturbation to Calabi-Yau metric
• 046W Proposition Proposition 4.37 .
• 046X Theorem Theorem 4.38 .
• 046Y Section 4.11. Ooguri-Vafa type metrics on the negative vertex
• 046Z Corollary Corollary 4.39 .
• 0470 Corollary Corollary 4.40 .
• 0471 Proposition Proposition 4.41 .
• 0472 Proof Proof.
• 0473 Section 4.12. Special Lagrangian geometry
• 0474 Section 4.13. Incompleteness and running coupling
• 0475 Remark Remark 4.9 .
• 0476 Remark Remark 4.10 .
• 0477 Section 1 Foreword
• 0478 Remark Remark 1.1 .
• 0479 Remark Remark 1.2 .
• 047A Remark Remark 1.3 .
• 047B Acknowledgement Acknowledgement .
• 047C Section 2 Thomas-Yau conjecture backgrounds
• 047D Subsection 2.1 The Thomas-Yau-Joyce picture
• 047E Subsubsection Thomas-Yau’s proposal
• 047F Lemma Lemma 2.1 .
• 047G Proof Proof.
• 047H Remark Remark 2.1 .
• 047I Conjecture Conjecture 2.2 .
• 047J Subsubsection Bridgeland stability, Joyce’s proposal
• 047K Definition Definition 2.3 .
• 047L Remark Remark 2.2 .
• 047M Remark Remark 2.3 .
• 047N Subsubsection Why Thomas-Yau has predictive power
• 047P Question Question 1 .
• 047Q Remark Remark 2.4 .
• 047R Subsection 2.2 Principal evidence of Thomas-Yau
• 047S Proposition Proposition 2.4 .
• 047T Proof Proof.
• 047U Remark Remark 2.5 .
• 047V Remark Remark 2.6 .
• 047W Theorem Theorem 2.5 .
• 047X Remark Remark 2.7 .
• 047Y Proof Proof.
• 047Z Subsection 2.3 Variant: uniqueness of the Lawlor neck
• 0480 Subsubsection Lawlor necks
• 0481 Subsubsection Joyce-Imagi-Santos uniqueness theorem
• 0482 Theorem Theorem 2.6 .
• 0483 Remark Remark 2.8 .
• 0484 Lemma Lemma 2.7 .
• 0485 Proof Proof.
• 0486 Subsubsection Ideal triangles
• 0487 Example Example 2.8 .
• 0488 Subsection 2.4 Philosophy of open string mirror symmetry
• 0489 Subsubsection Caveats for differential geometers
• 048A Subsection 2.5 Hermitian-Yang-Mills
• 048B Theorem Theorem 2.9 .
• 048C Remark Remark 2.9 .
• 048D Subsubsection μ -stability and its wider context
• 048E Remark Remark 2.10 .
• 048F Subsubsection Donaldson functional
• 048G Subsubsection Proof idea of Donaldson-Uhlenbeck-Yau theorem
• 048H Remark Remark 2.11 .
• 048I Remark Remark 2.12 .
• 048J Subsubsection Infinite dimensional GIT picture, and possible lack of mirror analgoue
• 048K Subsection 2.6 Deformed Hermitian Yang-Mills
• 048L Subsubsection Towards a Bridgeland stability condition
• 048M Subsubsection Main achievements on dHYM
• 048N Subsubsection Will dHYM lead to the mirror Bridgeland condition?
• 048P Subsection 2.7 Extensions and wall crossing
• 048Q Subsubsection Extension bundle vs. Lagrangian connection sum
• 048R Subsubsection Wall crossing
• 048S Subsection 2.8 Space of almost calibrated Lagrangians
• 048T Theorem Theorem 2.10 .
• 048U Theorem Theorem 2.11 .
• 048V Subsubsection Limitations
• 048W Subsubsection Exact isotopy class versus derived category class
• 048X Example Example 2.12 .
• 048Y Remark Remark 2.13 .
• 048Z Question Question 2 .
• 0490 Remark Remark 2.14 .
• 0491 Subsection 2.9 Totally real geometry
• 0492 Proposition Proposition 2.13 .
• 0493 Subsubsection Limitations
• 0494 Subsection 2.10 Analogy with tunneling effect
• 0495 Section 3 Moduli space of holomorphic curves
• 0496 Remark Remark 3.1 .
• 0497 Subsection 3.1 Lotay-Pacini picture revisited
• 0498 Subsubsection High brow viewpoint
• 0499 Subsubsection 3.1.1 The exact embedded Lagrangian case
• 049A Proposition Proposition 3.1 .
• 049B Remark Remark 3.2 .
• 049C Remark Remark 3.3 .
• 049D Subsubsection 3.1.2 Immersed case
• 049E Example Example 3.2 .
• 049F Remark Remark 3.4 .
• 049G Question Question 3 .
• 049H Question Question 4 .
• 049I Subsubsection Distinguished triangles
• 049J Subsection 3.2 Solomon functional revisited
• 049K Subsubsection Homological nature of the Solomon functional
• 049L Lemma Lemma 3.3 .
• 049M Proof Proof.
• 049N Subsubsection Proposed extension of the Solomon functional
• 049P Subsubsection Change of reference Lagrangian
• 049Q Proposition Proposition 3.4 .
• 049R Proof Proof.
• 049S Remark Remark 3.5 .
• 049T Subsection 3.3 Automatic transversality
• 049U Subsubsection Index theory preliminary
• 049V Subsubsection 3.3.1 Automatically transverse cases
• 049W Subsubsection Holomorphic strip
• 049X Lemma Lemma 3.5 .
• 049Y Proof Proof.
• 049Z Corollary Corollary 3.6 .
• 04A0 Proof Proof.
• 04A1 Subsubsection Holomorphic polygon
• 04A2 Lemma Lemma 3.7 .
• 04A3 Remark Remark 3.6 .
• 04A4 Proposition Proposition 3.8 .
• 04A5 Proof Proof.
• 04A6 Remark Remark 3.7 .
• 04A7 Subsubsection Weighted Sobolev space with exponential growth
• 04A8 Lemma Lemma 3.9 .
• 04A9 Proof Proof.
• 04AA Lemma Lemma 3.10 .
• 04AB Lemma Lemma 3.11 .
• 04AC Proof Proof.
• 04AD Subsubsection Structure of linearized Cauchy-Riemann equation
• 04AE Lemma Lemma 3.12 .
• 04AF Proof Proof.
• 04AG Lemma Lemma 3.13 .
• 04AH Proof Proof.
• 04AI Lemma Lemma 3.14 .
• 04AJ Corollary Corollary 3.15 .
• 04AK Proposition Proposition 3.16 .
• 04AL Corollary Corollary 3.17 .
• 04AM Corollary Corollary 3.18 .
• 04AN Corollary Corollary 3.19 .
• 04AP Subsubsection Hamiltonian deformations and transversality
• 04AQ Proposition Proposition 3.20 .
• 04AR Proof Proof.
• 04AS Remark Remark 3.8 .
• 04AT Subsubsection Further comments on automatic transversality
• 04AU Subsection 3.4 Positivity condition
• 04AV Question Question 5 .
• 04AW Subsubsection Morse theory analogy
• 04AX Subsubsection Positivity for individual moduli spaces
• 04AY Question Question 6 .
• 04AZ Question Question 7 .
• 04B0 Remark Remark 3.9 .
• 04B1 Subsection 3.5 Floer theoretic obstructions
• 04B2 Subsubsection General features of obstruction conditions
• 04B3 Subsubsection The Floer theoretic obstruction condition
• 04B4 Theorem Theorem 3.21 .
• 04B5 Proof Proof.
• 04B6 Claim Claim 3.22 .
• 04B7 Claim Claim 3.23 .
• 04B8 Claim Claim 3.24 .
• 04B9 Lemma Lemma 3.25 .
• 04BA Remark Remark 3.10 .
• 04BB Remark Remark 3.11 .
• 04BC Subsubsection Variant: twisted complex case
• 04BD Theorem Theorem 3.26 .
• 04BE Proof Proof.
• 04BF Lemma Lemma 3.27 .
• 04BG Proof Proof.
• 04BH Subsubsection What if we relax the positivity condition?
• 04BI Theorem Theorem 3.28 .
• 04BJ Proof Proof.
• 04BK Claim Claim 3.29 .
• 04BL Claim Claim 3.30 .
• 04BM Remark Remark 3.12 .
• 04BN Remark Remark 3.13 .
• 04BP Subsection 3.6 Towards a Bridgeland stability condition
• 04BQ Subsubsection Joyce’s proposal and Bridgeland stability condition revisited
• 04BR Conjecture Conjecture 3.31 .
• 04BS Proof Proof.
• 04BT Remark Remark 3.14 .
• 04BU Definition Definition 3.32 .
• 04BV Conjecture Conjecture 3.33 .
• 04BW Proof Proof.
• 04BX Conjecture Conjecture 3.34 .
• 04BY Proof Proof.
• 04BZ Remark Remark 3.15 .
• 04C0 Subsubsection Almost calibrated case: categorical predictions of the Joyce picture
• 04C1 Subsection 3.7 Moduli integral formula for the Solomon functional
• 04C2 Subsubsection 3.7.1 Moduli integral formula for the Solomon functional
• 04C3 Proposition Proposition 3.35 .
• 04C4 Remark Remark 3.16 .
• 04C5 Remark Remark 3.17 .
• 04C6 Subsubsection 3.7.2 Change of reference Lagrangians formula revisited
• 04C7 Claim Claim 3.36 .
• 04C8 Claim Claim 3.37 .
• 04C9 Claim Claim 3.38 .
• 04CA Subsubsection 3.7.3 First variation formula revisited
• 04CB Subsubsection 3.7.4 Speculations on compact Calabi-Yau manifolds
• 04CC Remark Remark 3.18 .
• 04CD Remark Remark 3.19 .
• 04CE Conjecture Conjecture 3.39 .
• 04CF Subsection 3.8 More applications of moduli space integrals
• 04CG Subsubsection 3.8.1 Lotay-Pacini convexity
• 04CH Subsubsection 3.8.2 Lower bound of the Solomon functional
• 04CI Proposition Proposition 3.40 .
• 04CJ Proof Proof.
• 04CK Remark Remark 3.20 .
• 04CL Subsubsection 3.8.3 Bounded part of the Solomon functional
• 04CM Proposition Proposition 3.41 .
• 04CN Proof Proof.
• 04CP Lemma Lemma 3.42 .
• 04CQ Proof Proof.
• 04CR Theorem Theorem 3.43 .
• 04CS Proof Proof.
• 04CT Remark Remark 3.21 .
• 04CU Remark Remark 3.22 .
• 04CV Remark Remark 3.23 .
• 04CW Subsubsection What if we relax the positivity condition?
• 04CX Section 4 Continuity, LMCF and variational method
• 04CY Subsection 4.1 Lagrangian mean curvature flow
• 04CZ Subsubsection LMCF basics
• 04D0 Proposition Proposition 4.1 .
• 04D1 Remark Remark 4.1 .
• 04D2 Subsubsection Finite time singularity, and prototypical bad behaviours
• 04D3 Example Example 4.2 .
• 04D4 Example Example 4.3 .
• 04D5 Example Example 4.4 .
• 04D6 Subsubsection Joyce’s LMCF proposal
• 04D7 Example Example 4.5 .
• 04D8 Example Example 4.6 .
• 04D9 Example Example 4.7 .
• 04DA Subsubsection Infinite time limit and its difficulties
• 04DB Subsection 4.2 Continuity method
• 04DC Subsubsection The continuity path
• 04DD Subsubsection Two main obstacles
• 04DE Subsubsection How to find an initial special Lagrangian
• 04DF Question Question 8 .
• 04DG Remark Remark 4.2 .
• 04DH Subsubsection Compactness and genericity
• 04DI Question Question 9 .
• 04DJ Example Example 4.8 .
• 04DK Subsubsection Wall crossing
• 04DL Remark Remark 4.3 .
• 04DM Subsubsection What’s the role of the brane structure?
• 04DN Subsubsection Comparison with LMCF
• 04DP Section 5 Variational method
• 04DQ Question Question 10 .
• 04DR Remark Remark 5.1 .
• 04DS Conjecture Conjecture 5.1 .
• 04DT Subsection 5.1 Compactness and regularity
• 04DU Subsubsection 5.1.1 Standard geometric measure theory
• 04DV Theorem Theorem 5.2 .
• 04DW Remark Remark 5.2 .
• 04DX Remark Remark 5.3 .
• 04DY Theorem Theorem 5.3 .
• 04DZ Remark Remark 5.4 .
• 04E0 Example Example 5.4 .
• 04E1 Theorem Theorem 5.5 .
• 04E2 Remark Remark 5.5 .
• 04E3 Remark Remark 5.6 .
• 04E4 Subsubsection 5.1.2 Exact Lagrangians under weak regularity
• 04E5 Lemma Lemma 5.6 .
• 04E6 Remark Remark 5.7 .
• 04E7 Lemma Lemma 5.7 .
• 04E8 Proof Proof.
• 04E9 Subsubsection Continuity of the Solomon functional
• 04EA Lemma Lemma 5.8 .
• 04EB Proof Proof.
• 04EC Subsubsection Robustness of potential clustering
• 04ED Remark Remark 5.8 .
• 04EE Corollary Corollary 5.9 .
• 04EF Proof Proof.
• 04EG Subsubsection 5.1.3 Almgren’s regularity in the almost Calabi-Yau setting?
• 04EH Question Question 11 .
• 04EI Subsection 5.2 Quantitative almost calibratedness
• 04EJ Remark Remark 5.9 .
• 04EK Remark Remark 5.10 .
• 04EL Subsubsection Isoperimetric inequality
• 04EM Lemma Lemma 5.10 .
• 04EN Proof Proof.
• 04EP Remark Remark 5.11 .
• 04EQ Proposition Proposition 5.11 .
• 04ER Proof Proof.
• 04ES Subsubsection Volume monotonicity and lower bound
• 04ET Corollary Corollary 5.12 .
• 04EU Proof Proof.
• 04EV Remark Remark 5.12 .
• 04EW Subsubsection No escape to spatial infinity
• 04EX Corollary Corollary 5.13 .
• 04EY Proof Proof.
• 04EZ Subsubsection Nontriviality of homology classes
• 04F0 Corollary Corollary 5.14 .
• 04F1 Proof Proof.
• 04F2 Subsubsection 5.2.1 Intrinsic distance bound and potential clustering
• 04F3 Lemma Lemma 5.15 .
• 04F4 Corollary Corollary 5.16 .
• 04F5 Proof Proof.
• 04F6 Corollary Corollary 5.17 .
• 04F7 Proof Proof.
• 04F8 Corollary Corollary 5.18 .
• 04F9 Proof Proof.
• 04FA Remark Remark 5.13 .
• 04FB Subsubsection 5.2.2 Bounded part of the Solomon functional revisited
• 04FC Proposition Proposition 5.19 .
• 04FD Proof Proof.
• 04FE Subsection 5.3 Variational strategy
• 04FF Remark Remark 5.14 .
• 04FG Subsubsection L p -Smoothing property and Joyce’s LMCF
• 04FH Conjecture Conjecture 5.20 .
• 04FI Remark Remark 5.15 .
• 04FJ Remark Remark 5.16 .
• 04FK Subsection 5.4 Floer theory under weak regularity
• 04FL Remark Remark 5.17 .
• 04FM Subsubsection Floer theoretic difficulties
• 04FN Subsubsection Formal limit perspective
• 04FP Question Question 12 .
• 04FQ Remark Remark 5.18 .
• 04FR Question Question 13 .
• 04FS Subsubsection Geometric perspective: bordism currents and triangulated categories
• 04FT Question Question 14 .
• 04FU Remark Remark 5.19 .
• 04FV Remark Remark 5.20 .
• 04FW Question Question 15 .
• 04FX Subsubsection Previlleged role of H ​ F 0
• 04FY Remark Remark 5.21 .
• 04FZ Remark Remark 5.22 .
• 04G0 Subsubsection Multiplicity issues
• 04G1 Remark Remark 5.23 .
• 04G2 Remark Remark 5.24 .
• 04G3 Remark Remark 5.25 .
• 04G4 Subsection 5.5 Asymptotes of the Solomon functional
• 04G5 Remark Remark 5.26 .
• 04G6 Subsubsection 5.5.1 Thomas-Yau conjecture
• 04G7 Conjecture Conjecture 5.21 .
• 04G8 Conjecture Conjecture 5.22 .
• 04G9 Proof Proof.
• 04GA Remark Remark 5.27 .
• 04GB Remark Remark 5.28 .
• 04GC Subsection 5.6 Minimizers and special Lagrangians
• 04GD Conjecture Conjecture 5.23 .
• 04GE Subsubsection LMCF viewpoint
• 04GF Subsubsection Hamiltonian variations
• 04GG Remark Remark 5.29 .
• 04GH Subsection 5.7 Thomas-Yau uniqueness revisited
• 04GI Conjecture Conjecture 5.24 .
• 04GJ Proof Proof.
• 04GK Question Question 16 .
• 04GL Subsubsection 5.7.1 Special Lagrangians are minimizers
• 04GM Conjecture Conjecture 5.25 .
• 04GN Proof Proof.
• 04GP Subsection 5.8 Comparison with Joyce’s LMCF program
• 04GQ Section 6 Appendix on the Fukaya category
• 04GR Subsection 6.1 Fukaya category for embedded exact Lagrangians
• 04GS Subsubsection Floer cohomology and A ∞ -structure with mod 2 coefficients
• 04GT Remark Remark 6.1 .
• 04GU Remark Remark 6.2 .
• 04GV Remark Remark 6.3 .
• 04GW Remark Remark 6.4 .
• 04GX Remark Remark 6.5 .
• 04GY Remark Remark 6.6 .
• 04GZ Remark Remark 6.7 .
• 04H0 Subsubsection Self Floer cohomology
• 04H1 Example Example 6.1 .
• 04H2 Subsubsection Sign issues and brane structures
• 04H3 Remark Remark 6.8 .
• 04H4 Remark Remark 6.9 .
• 04H5 Example Example 6.2 .
• 04H6 Subsubsection Twisted complexes, distinguished triangles, derived category
• 04H7 Remark Remark 6.10 .
• 04H8 Subsection 6.2 Immersed exact Lagrangians
• 04H9 Subsubsection Teardrop curves and obstructions
• 04HA Remark Remark 6.11 .
• 04HB Remark Remark 6.12 .
• 04HC Subsubsection Cancellation of obstructions
• 04HD Remark Remark 6.13 .
• 04HE Subsubsection The union of several components
• 04HF Lemma Lemma 6.3 .
• 04HG Proof Proof.
• 04HH Subsubsection Orientation signs on bordism currents
• 04HI Section 1 Introduction.
• 04HJ Section 2 The topology.
• 04HK Definition Definition 2.1 .
• 04HL Assumption Assumption 2.2 .
• 04HM Definition Definition 2.3 .
• 04HN Proposition Proposition 2.4 .
• 04HP Remark Remark 2.5 .
• 04HQ Example Example 2.6 (Nodal fibration) .
• 04HR Example Example 2.7 (Generic singular fibration) .
• 04HS Example Example 2.8 (Negative fibration) .
• 04HT Example Example 2.9 (Alternative negative fibration) .
• 04HU Example Example 2.10 (Positive fibration) .
• 04HV Theorem Theorem 2.11 (Gross) .
• 04HW Section 3 Affine manifolds and Lagrangian fibrations
• 04HX Definition Definition 3.1 .
• 04HY Subsection Action-angle coordinates.
• 04HZ Proposition Proposition 3.2 (Arnold-Liouville) .
• 04I0 Proof Proof.
• 04I1 Theorem Theorem 3.3 (Duistermaat) .
• 04I2 Proof Proof.
• 04I3 Corollary Corollary 3.4 .
• 04I4 Proof Proof.
• 04I5 Corollary Corollary 3.5 .
• 04I6 Subsection Affine manifolds with singularities.
• 04I7 Definition Definition 3.6 .
• 04I8 Example Example 3.7 (The node) .
• 04I9 Example Example 3.8 (The edge) .
• 04IA Example Example 3.9 (A variation) .
• 04IB Example Example 3.10 (Positive vertex) .
• 04IC Example Example 3.11 (A variation) .
• 04ID Example Example 3.12 (Negative vertex) .
• 04IE Example Example 3.13 (A variation) .
• 04IF Definition Definition 3.14 .
• 04IG Corollary Corollary 3.15 .
• 04IH Subsection Examples
• 04II Example Example 3.16 .
• 04IJ Example Example 3.17 .
• 04IK Example Example 3.18 (A variation) .
• 04IL Theorem Theorem 3.19 (Gross [ 7 ] ) .
• 04IM Subsection The focus-focus fibration
• 04IN Example Example 3.20 .
• 04IP Remark Remark 3.21 .
• 04IQ Subsection The K3 surface.
• 04IR Theorem Theorem 3.22 .
• 04IS Proof Proof.
• 04IT Corollary Corollary 3.23 .
• 04IU Section 4 Positive and generic-singular fibrations.
• 04IV Definition Definition 4.1 .
• 04IW Subsection Examples
• 04IX Example Example 4.2 .
• 04IY Example Example 4.3 .
• 04IZ Example Example 4.4 .
• 04J0 Example Example 4.5 .
• 04J1 Subsection The affine structures.
• 04J2 Theorem Theorem 4.6 .
• 04J3 Remark Remark 4.7 .
• 04J4 Proposition Proposition 4.8 .
• 04J5 Proof Proof.
• 04J6 Corollary Corollary 4.9 .
• 04J7 Proof Proof.
• 04J8 Proposition Proposition 4.10 .
• 04J9 Proposition Proposition 4.11 .
• 04JA Proof Proof.
• 04JB Subsection Gluing over the discriminant locus
• 04JC Remark Remark 4.12 .
• 04JD Theorem Theorem 4.13 .
• 04JE Lemma Lemma 4.14 .
• 04JF Proof Proof.
• 04JG Lemma Lemma 4.15 .
• 04JH Corollary Corollary 4.16 .
• 04JI Proposition Proposition 4.17 .
• 04JJ Proof Proof.
• 04JK Subsection Gluing legs
• 04JL Proposition Proposition 4.18 .
• 04JM Theorem Theorem 4.19 .
• 04JN Section 5 Piecewise smooth fibrations
• 04JP Subsection Fibrations with torus symmetry.
• 04JQ Remark Remark 5.1 .
• 04JR Proposition Proposition 5.2 .
• 04JS Remark Remark 5.3 .
• 04JT Subsection The reduced geometry.
• 04JU Subsection A construction
• 04JV Proposition Proposition 5.4 .
• 04JW Subsection Examples
• 04JX Example Example 5.5 (The amoeba) .
• 04JY Example Example 5.6 (Stitched focus-focus) .
• 04JZ Example Example 5.7 (The leg) .
• 04K0 Example Example 5.8 (The amoeba with thin legs) .
• 04K1 Proposition Proposition 5.9 .
• 04K2 Proof Proof.
• 04K3 Section 6 Stitched fibrations
• 04K4 Definition Definition 6.1 .
• 04K5 Example Example 6.2 (Stitched focus-focus, revisited) .
• 04K6 Example Example 6.3 (The amoeba, revisited) .
• 04K7 Definition Definition 6.4 .
• 04K8 Proposition Proposition 6.5 .
• 04K9 Proof Proof.
• 04KA Definition Definition 6.6 .
• 04KB Example Example 6.7 (Normal forms) .
• 04KC Definition Definition 6.8 .
• 04KD Proposition Proposition 6.9 .
• 04KE Proof Proof.
• 04KF Definition Definition 6.10 .
• 04KG Theorem Theorem 6.11 .
• 04KH Proof Proof.
• 04KI Theorem Theorem 6.12 .
• 04KJ Example Example 6.13 .
• 04KK Subsection Monodromy
• 04KL Example Example 6.14 .
• 04KM Theorem Theorem 6.15 .
• 04KN Proof Proof.
• 04KP Remark Remark 6.16 .
• 04KQ Example Example 6.17 .
• 04KR Example Example 6.18 .
• 04KS Theorem Theorem 6.19 .
• 04KT Remark Remark 6.20 .
• 04KU Example Example 6.21 .
• 04KV Subsection Non-proper stitched fibrations
• 04KW Assumption Assumption 6.22 .
• 04KX Proposition Proposition 6.23 .
• 04KY Proof Proof.
• 04KZ Remark Remark 6.24 .
• 04L0 Example Example 6.25 (Normal form of cylindrical type) .
• 04L1 Definition Definition 6.26 .
• 04L2 Remark Remark 6.27 .
• 04L3 Proposition Proposition 6.28 .
• 04L4 Proof Proof.
• 04L5 Theorem Theorem 6.29 .
• 04L6 Proposition Proposition 6.30 .
• 04L7 Section 7 Lagrangian negative fibrations
• 04L8 Definition Definition 7.1 .
• 04L9 Proposition Proposition 7.2 .
• 04LA Theorem Theorem 7.3 .
• 04LB Subsection Smoothing I
• 04LC Lemma Lemma 7.4 .
• 04LD Proof Proof.
• 04LE Remark Remark 7.5 .
• 04LF Subsection Smoothing II
• 04LG Lemma Lemma 7.6 .
• 04LH Proof Proof.
• 04LI Remark Remark 7.7 .
• 04LJ Subsection The normal form
• 04LK Lemma Lemma 7.8 .
• 04LL Proof Proof.
• 04LM Lemma Lemma 7.9 .
• 04LN Proof Proof.
• 04LP Lemma Lemma 7.10 .
• 04LQ Proof Proof.
• 04LR Corollary Corollary 7.11 .
• 04LS Proof Proof.
• 04LT Subsection Smoothing III
• 04LU Lemma Lemma 7.12 .
• 04LV Proof Proof.
• 04LW Proof Proof of Theorem 7.3 .
• 04LX Section 8 The compactification.
• 04LY Subsection The main theorem
• 04LZ Definition Definition 8.1 .
• 04M0 Theorem Theorem 8.2 .
• 04M1 Proof Proof.
• 04M2 Corollary Corollary 8.3 .
• 04M3 Proof Proof.
• 04M4 Section Introduction
• 04M5 Theorem Theorem A.
• 04M6 Theorem Theorem B.
• 04M7 Corollary Corollary C.
• 04M8 Section 1 Preliminaries
• 04M9 Subsection 1.1 Models
• 04MA Definition Definition 1.1.1 .
• 04MB Subsection 1.2 Toric geometry
• 04MC Lemma Lemma 1.2.1 ( [ Ful93 , 3.4] ).
• 04MD Corollary Corollary 1.2.2.
• 04ME Lemma Lemma 1.2.3 ( [ Ful93 , p. 99] ).
• 04MF Proposition Proposition 1.2.5.
• 04MG Proof Proof.
• 04MH Definition Definition 1.2.6 .
• 04MI Subsection 1.3 Berkovich spaces
• 04MJ Subsection 1.4 Skeletons
• 04MK Definition Definition 1.4.1 .
• 04ML Definition Definition 1.4.2 .
• 04MM Proposition Proposition 1.4.3 ( [ MN15 , Proposition 2.4.4] ).
• 04MN Definition Definition 1.4.4 .
• 04MP Definition Definition 1.4.5 .
• 04MQ Subsection 1.5 Berkovich retractions
• 04MR Definition Definition 1.5.1 .
• 04MS Proposition Proposition 1.5.2 ( [ NXY19 , Example 3.5] ).
• 04MT Proof Proof.
• 04MU Lemma Lemma 1.5.3.
• 04MV Proof Proof.
• 04MW Subsection 1.6 Affinoid torus fibrations and integral affine structures
• 04MX Definition Definition 1.6.1 .
• 04MY Example Example 1.6.2 .
• 04MZ Example Example 1.6.3 .
• 04N0 Definition Definition 1.6.4 .
• 04N1 Definition Definition 1.6.5 .
• 04N2 Lemma Lemma 1.6.6 ( [ KS06 , 2.1] ).
• 04N3 Remark Remark 1.6.7 .
• 04N4 Subsection 1.7 The Calabi–Yau case
• 04N5 Definition Definition 1.7.1 .
• 04N6 Theorem Theorem 1.7.2 ( [ NXY19 , Theorem 1.13] ).
• 04N7 Definition Definition 1.7.3 .
• 04N8 Definition Definition 1.7.4 .
• 04N9 Example Example 1.7.5 .
• 04NA Theorem Theorem 1.7.6 ( [ NXY19 , Theorem 6.1] ).
• 04NB Example Example 1.7.7 .
• 04NC Section 2 Toric structure along toric strata
• 04ND Theorem Theorem B .
• 04NE Corollary Corollary C .
• 04NF Proof Proof.
• 04NG Subsection 2.1 Notation and strategy
• 04NH Remark Remark 2.1.1 .
• 04NI Remark Remark 2.1.2 .
• 04NJ Lemma Lemma 2.1.3.
• 04NK Proof Proof.
• 04NL Proposition Proposition 2.1.6.
• 04NM Proof Proof.
• 04NN Subsection 2.2 Construction of the divisors
• 04NP Lemma Lemma 2.2.1.
• 04NQ Proof Proof.
• 04NR Lemma Lemma 2.2.3.
• 04NS Proof Proof.
• 04NT Subsection 2.3 Construction of the sections for a maximal cone
• 04NU Lemma Lemma 2.3.1.
• 04NV Proof Proof.
• 04NW Subsection 2.4 Construction for two adjacent maximal cones
• 04NX Lemma Lemma 2.4.2.
• 04NY Proof Proof.
• 04NZ Subsection 2.5 Construction of the morphism
• 04P0 Lemma Lemma 2.5.1.
• 04P1 Proof Proof.
• 04P2 Lemma Lemma 2.5.2.
• 04P3 Proof Proof.
• 04P4 Proposition Proposition 2.5.3.
• 04P5 Proof Proof.
• 04P6 Subsection 2.6 Integral affine structure and toric irreducible components
• 04P7 Corollary Corollary 2.6.1.
• 04P8 Proof Proof.
• 04P9 Section 3 Integral affine structures
• 04PA Subsection 3.1 Integral affine structure induced by a model
• 04PB Proposition Proposition 3.1.1.
• 04PC Proof Proof.
• 04PD Lemma Lemma 3.1.4.
• 04PE Proof Proof.
• 04PF Remark Remark 3.1.5 .
• 04PG Subsubsection 3.1.1 Case of K3 surfaces
• 04PH Corollary Corollary 3.1.6.
• 04PI Proof Proof.
• 04PJ Remark Remark 3.1.7 .
• 04PK Subsection 3.2 Integral affine structure induced by combining several models
• 04PL Definition Definition 3.2.1 .
• 04PM Proposition Proposition 3.2.2.
• 04PN Proof Proof.
• 04PP Subsubsection 3.2.1 Case of K3 surfaces
• 04PQ Corollary Corollary 3.2.4.
• 04PR Subsection 3.3 Degeneration of quartic K3 surfaces
• 04PS Subsubsection 3.3.1 Integral affine structure induced by the model 𝒳 i ​ j ​ k
• 04PT Subsubsection 3.3.2 Integral affine structure induced combining more models
• 04PU Lemma Lemma 3.3.2.
• 04PV Proof Proof.
• 04PW Subsubsection 3.3.3 Dispersion of singularities
• 04PX Proposition Proposition 3.3.3.
• 04PY Proof Proof.
• 04PZ Subsubsection 3.3.4 Collision of singularities
• 04Q0 Section 4 Degeneration of quintic 3-folds
• 04Q1 Subsection 4.1 Setting and plan of the proof
• 04Q2 Theorem Theorem A .
• 04Q3 Remark Remark 4.1.1 .
• 04Q4 Subsection 4.2 Local resolution
• 04Q5 Subsection 4.3 Local dominating model
• 04Q6 Subsection 4.4 Local combinatorial retraction
• 04Q7 Subsection 4.5 Minimal models 𝒳 i ​ j ​ k ​ l
• 04Q8 Subsection 4.6 Dominating model and combinatorial retraction
• 04Q9 Subsection 4.7 Monodromy representation
• 04QA Remark Remark 4.7.1 .
• 04QB Remark Remark 4.7.2 .
• 04QC Subsection 4.8 Comparison to Gromov-Hausdorff limit of Fermat families
• 04QD Proposition Proposition 4.8.1.
• 04QE Proof Proof.
• 04QF Section 5 Appendix
• 04QG Definition Definition 5.0.1 .
• 04QH Definition Definition 5.0.2 .
• 04QI Lemma Lemma 5.0.3.
• 04QJ Proof Proof.
• 04QK Proposition Proposition 5.0.4.
• 04QL Proof Proof.
• 04QM Acknowledgement Acknowledgement .
• 04QN Section 1. Preliminaries
• 04QP Subsection 1.1. Balanced polyhedra
• 04QQ Definition Definition 1 .
• 04QR Definition Definition 2 .
• 04QS Definition Definition 3 .
• 04QT Example Example 1 .
• 04QU Example Example 2 .
• 04QV Proposition Proposition 1.1 .
• 04QW Proof Proof.
• 04QX Proposition Proposition 1.2 .
• 04QY Proof Proof.
• 04QZ Remark Remark 1.3 .
• 04R0 Proposition Proposition 1.4 .
• 04R1 Proof Proof.
• 04R2 Corollary Corollary 1.5 .
• 04R3 Corollary Corollary 1.6 .
• 04R4 Proof Proof.
• 04R5 Subsection 1.2. Maximal polyhedral complexes and their decomposition into primitive pieces
• 04R6 Definition Definition 4 .
• 04R7 Proposition Proposition 1.7 .
• 04R8 Proof Proof.
• 04R9 Example Example 3 .
• 04RA Proposition Proposition 1.8 .
• 04RB Proof Proof.
• 04RC Remark Remark 1.9 .
• 04RD Proposition Proposition 1.10 .
• 04RE Proof Proof.
• 04RF Definition Definition 5 .
• 04RG Proposition Proposition 1.11 .
• 04RH Proof Proof.
• 04RI Subsection 1.3. Toric varieties and compactification of balanced polyhedra
• 04RJ Definition Definition 6 .
• 04RK Proposition Proposition 1.12 .
• 04RL Proposition Proposition 1.13 .
• 04RM Proof Proof.
• 04RN Proposition Proposition 1.14 .
• 04RP Remark Remark 1.15 .
• 04RQ Subsection 1.4. Stratified fibrations
• 04RR Definition Definition 7 .
• 04RS Proposition Proposition 1.16 .
• 04RT Remark Remark 1.17 .
• 04RU Subsection 1.5. Hypersurfaces in toric varieties
• 04RV Proposition Proposition 1.18 .
• 04RW Remark Remark 1.19 .
• 04RX Proof Proof.
• 04RY Lemma Lemma 1.20 .
• 04RZ Proof Proof.
• 04S0 Remark Remark 1.21 .
• 04S1 Example Example 4 .
• 04S2 Example Example 5 .
• 04S3 Subsection 1.6. Pairs-of-pants in higher dimensions
• 04S4 Definition Definition 8 .
• 04S5 Proposition Proposition 1.22 .
• 04S6 Remark Remark 1.23 .
• 04S7 Proposition Proposition 1.24 .
• 04S8 Proof Proof.
• 04S9 Section 2. Statement of the results
• 04SA Theorem Theorem 1 .
• 04SB Theorem Theorem 1’ .
• 04SC Theorem Theorem 2 .
• 04SD Theorem Theorem 3 .
• 04SE Remark Remark 2.1 .
• 04SF Section 3. Some examples
• 04SG Subsection 3.1. Riemann surfaces
• 04SH Subsection 3.2. The elliptic curve and the K3-surface
• 04SI Subsection 3.3. Hyperplanes in the projective space
• 04SJ Lemma Lemma 3.1 .
• 04SK Proof Proof.
• 04SL Lemma Lemma 3.2 .
• 04SM Lemma Lemma 3.3 (cf. Lemma 3 of [ 11 ] ) .
• 04SN Proof Proof.
• 04SP Corollary Corollary 3.4 .
• 04SQ Proof Proof.
• 04SR Corollary Corollary 3.5 .
• 04SS Proof Proof.
• 04ST Subsection 3.4. A localization Q n ⊂ ( ℂ ∗ ) n + 1 of the standard hyperplane
• 04SU Proposition Proposition 3.6 .
• 04SV Proof Proof.
• 04SW Section 4. Reconstruction of the complex hypersurface from a balanced polyhedron Π
• 04SX Theorem Theorem 4 .
• 04SY Corollary Corollary 4.1 .
• 04SZ Remark Remark 4.2 .
• 04T0 Section 5. Proof of the main theorems
• 04T1 Subsection 5.1. Viro’s patchworking
• 04T2 Remark Remark 5.1 .
• 04T3 Subsection 5.2. Non-Archimedian amoebas
• 04T4 Theorem Theorem (Kapranov [ 7 ] ) .
• 04T5 Subsection 5.3. Lifts of non-Archimedian amoebas to ( ℂ ∗ ) n + 1
• 04T6 Lemma Lemma 5.2 .
• 04T7 Proof Proof.
• 04T8 Subsection 5.4. Maslov’s dequantization
• 04T9 Lemma Lemma 5.3 .
• 04TA Proof Proof.
• 04TB Corollary Corollary 5.4 .
• 04TC Proof Proof.
• 04TD Theorem Theorem 5 .
• 04TE Subsection 5.5. Construction of the fibration λ t : V t ∘ → Π
• 04TF Subsection 5.6. Proof of Theorems 2 and 4
• 04TG Lemma Lemma 5.5 .
• 04TH Proof Proof.
• 04TI Subsection 5.7. Proof of Theorems 1 , 1’ and 3
• 04TJ Section 1. Introduction
• 04TK Theorem Theorem 1.1 .
• 04TL Theorem Theorem 1.2 .
• 04TM Theorem Theorem 1.3 .
• 04TN Section 2. Preliminaries
• 04TP Lemma Lemma 2.1 .
• 04TQ Lemma Lemma 2.2 .
• 04TR Proof Proof.
• 04TS Lemma Lemma 2.3 .
• 04TT Proof Proof.
• 04TU Lemma Lemma 2.4 .
• 04TV Proof Proof.
• 04TW Section 3. Proof of Theorem 1.1
• 04TX Theorem Theorem 3.1 .
• 04TY Proof Proof of Theorem 1.1 : .
• 04TZ Lemma Lemma 3.2 .
• 04U0 Proof Proof.
• 04U1 Lemma Lemma 3.3 .
• 04U2 Proof Proof.
• 04U3 Proof Proof of Theorem 3.1 : .
• 04U4 Remark Remark 3.4 .
• 04U5 Section 4. Examples
• 04U6 Lemma Lemma 4.1 .
• 04U7 Remark Remark 4.2 .
• 04U8 Section 5. W 2 , 1 Regularity
• 04U9 Theorem Theorem 5.1 .
• 04UA Proof Proof of Theorem 1.2 : .
• 04UB Remark Remark 5.2 .
• 04UC Section 6. Unique Continuation
• 04UD Theorem Theorem 6.1 .
• 04UE Theorem Theorem 6.2 .
• 04UF Proof Proof of Theorem 1.3 : .
• 04UG Section 1. Introduction
• 04UH Subsection Acknowledgements
• 04UI Subsection Terminology and conventions
• 04UJ Section 2. Minimal d ​ l ​ t -models
• 04UK Subsection 2.1. Models and log pullbacks
• 04UL Subsubsection Subsubsection
• 04UM Subsubsection Subsubsection
• 04UN Subsubsection Subsubsection
• 04UP Subsubsection Subsubsection
• 04UQ Subsubsection Subsubsection
• 04UR Subsubsection Subsubsection
• 04US Subsection 2.2. d ​ l ​ t -models
• 04UT Subsubsection Subsubsection
• 04UU Subsubsection Subsubsection
• 04UV Subsubsection Subsubsection
• 04UW Subsubsection Subsubsection
• 04UX Subsubsection Subsubsection
• 04UY Theorem Theorem 2.2.6 .
• 04UZ Proof Proof.
• 04V0 Section 3. The essential skeleton
• 04V1 Subsection 3.1. Retraction to the skeleton of an s ​ n ​ c -model
• 04V2 Subsubsection Subsubsection
• 04V3 Subsubsection Subsubsection
• 04V4 Theorem Theorem 3.1.3 .
• 04V5 Proof Proof.
• 04V6 Subsubsection Subsubsection
• 04V7 Subsection 3.2. The skeleton of a good minimal d ​ l ​ t -model
• 04V8 Subsubsection Subsubsection
• 04V9 Subsubsection Subsubsection
• 04VA Lemma Lemma 3.2.3 .
• 04VB Proof Proof.
• 04VC Proposition Proposition 3.2.4 .
• 04VD Proof Proof.
• 04VE Corollary Corollary 3.2.5 .
• 04VF Proof Proof.
• 04VG Subsubsection Subsubsection
• 04VH Corollary Corollary 3.2.7 .
• 04VI Proof Proof.
• 04VJ Theorem Theorem 3.2.8 .
• 04VK Proof Proof.
• 04VL Corollary Corollary 3.2.9 .
• 04VM Proof Proof.
• 04VN Subsection 3.3. Kontsevich-Soibelman skeleta
• 04VP Subsubsection Subsubsection
• 04VQ Proposition Proposition 3.3.2 .
• 04VR Proof Proof.
• 04VS Theorem Theorem 3.3.4 .
• 04VT Proof Proof.
• 04VU Corollary Corollary 3.3.6 .
• 04VV Proof Proof.
• 04VW Section 4. The essential skeleton of a Calabi-Yau variety
• 04VX Subsection 4.1. The skeleton is a pseudo-manifold
• 04VY Subsubsection Subsubsection
• 04VZ Subsubsection Subsubsection
• 04W0 Subsubsection Subsubsection
• 04W1 Theorem Theorem 4.1.4 .
• 04W2 Proof Proof.
• 04W3 Subsubsection Subsubsection
• 04W4 Lemma Lemma 4.1.6 .
• 04W5 Proof Proof.
• 04W6 Theorem Theorem 4.1.7 .
• 04W7 Proof Proof.
• 04W8 Subsubsection Subsubsection
• 04W9 Lemma Lemma 4.1.9 .
• 04WA Proof Proof.
• 04WB Theorem Theorem 4.1.10 .
• 04WC Proof Proof.
• 04WD Subsection 4.2. Removing the algebraicity condition
• 04WE Subsubsection Subsubsection
• 04WF Proposition Proposition 4.2.2 .
• 04WG Proof Proof.
• 04WH Proposition Proposition 4.2.3 .
• 04WI Proof Proof.
• 04WJ Theorem Theorem 4.2.4 .
• 04WK Proof Proof.
• 04WL Section 1. Introduction
• 04WM Subsection Subsection
• 04WN Subsection Subsection
• 04WP Subsection Subsection
• 04WQ Subsection Subsection
• 04WR Subsection Preliminaries and notation
• 04WS Subsection Subsection
• 04WT Subsection Subsection
• 04WU Subsection Subsection
• 04WV Subsection Subsection
• 04WW Subsection Subsection
• 04WX Subsection Subsection
• 04WY Theorem Theorem 1.11 .
• 04WZ Proof Proof.
• 04X0 Subsection Subsection
• 04X1 Section 2. Construction of the non-archimedean SYZ fibration
• 04X2 Subsection Subsection
• 04X3 Subsection Subsection
• 04X4 Subsection Subsection
• 04X5 Subsection Subsection
• 04X6 Definition Definition 2.5 .
• 04X7 Subsection Subsection
• 04X8 Example Example 2.7 .
• 04X9 Proposition Proposition 2.8 .
• 04XA Proof Proof.
• 04XB Subsection Subsection
• 04XC Section 3. Affinoid torus fibrations
• 04XD Subsection Subsection
• 04XE Subsection Subsection
• 04XF Subsection Subsection
• 04XG Subsection Subsection
• 04XH Example Example 3.5 .
• 04XI Subsection Subsection
• 04XJ Subsection Subsection
• 04XK Proposition Proposition 3.8 .
• 04XL Proof Proof.
• 04XM Remark Remark 3.9 .
• 04XN Section 4. One-dimensional strata of minimal dlt-models
• 04XP Subsection Subsection
• 04XQ Lemma Lemma 4.2 .
• 04XR Proof Proof.
• 04XS Proposition Proposition 4.3 .
• 04XT Proof Proof.
• 04XU Corollary Corollary 4.4 .
• 04XV Proof Proof.
• 04XW Theorem Theorem 4.5 .
• 04XX Proof Proof.
• 04XY Corollary Corollary 4.6 .
• 04XZ Proof Proof.
• 04Y0 Section 5. Toric structure of snc-models along one-dimensional strata
• 04Y1 Subsection Subsection
• 04Y2 Subsection Subsection
• 04Y3 Proposition Proposition 5.4 .
• 04Y4 Proof Proof.
• 04Y5 Section 6. The smooth locus of the SYZ fibration
• 04Y6 Theorem Theorem 6.1 .
• 04Y7 Proof Proof.
• 04Y8 Subsection Subsection
• 04Y9 Section 1. Introduction
• 04YA Theorem Theorem 1.1 ( [ Od16 ] ) .
• 04YB Section 2. General Hermitian symmetric domain
• 04YC Theorem Theorem 2.1 .
• 04YD Proposition Proposition 2.2 .
• 04YE Corollary Corollary 2.3 (corollary to Theorem 2.1 and Proposition 2.2 ) .
• 04YF Remark Remark 2.4 .
• 04YG Section 3. Abelian varieties case
• 04YH Theorem Theorem 3.1 .
• 04YI Theorem Theorem 3.2 .
• 04YJ Section 4. Moduli of Algebraic K3 surfaces
• 04YK Subsection 4.1. Satake compactification
• 04YL Subsection 4.2. Tropical K3 surfaces
• 04YM Remark Remark 4.1 .
• 04YN Remark Remark 4.2 .
• 04YP Subsection 4.3. Gromov-Hausdorff collapse of K3 surfaces
• 04YQ Conjecture Conjecture 4.3 .
• 04YR Theorem Theorem 4.4 .
• 04YS Section 5. Moduli of Kähler K3 surfaces
• 04YT Theorem Theorem 5.1 .
• 04YU Section 6. Higher dimensional case
• 04YV Conjecture Conjecture 6.1 .
• 04YW Remark Remark 6.2 (Calabi-Yau case) .
• 04YX Section 1. Introduction
• 04YY Subsection 1.1. Background and main results
• 04YZ Theorem Theorem 1.1 .
• 04Z0 Remark Remark 1.1.1 .
• 04Z1 Remark Remark 1.1.2 .
• 04Z2 Remark Remark 1.1.3 .
• 04Z3 Remark Remark 1.1.4 .
• 04Z4 Subsection 1.2. Outline of the proof and organization of the paper
• 04Z5 Subsection 1.3. Acknowledgements
• 04Z6 Section 2. Calabi-Yau metrics with torus symmetry
• 04Z7 Subsection 2.1. Motivation and dimension reduction of the Calabi-Yau equation
• 04Z8 Subsection 2.2. Calabi model spaces
• 04Z9 Subsection 2.3. Two dimensional standard model spaces
• 04ZA Subsection 2.4. Linearized equation and singularities
• 04ZB Subsection 2.5. Higher rank torus symmetry
• 04ZC Section 3. Green’s currents
• 04ZD Subsection 3.1. Normal coordinates for an embedded submanifold
• 04ZE Definition Definition 3.1 (Normal coordinates) .
• 04ZF Lemma Lemma 3.2 (Generalized Gauss Lemma) .
• 04ZG Definition Definition 3.3 (Normal regularity order) .
• 04ZH Lemma Lemma 3.4 .
• 04ZI Proof Proof.
• 04ZJ Subsection 3.2. Green’s currents for Riemannian submanifolds
• 04ZK Definition Definition 3.5 ( k -current) .
• 04ZL Definition Definition 3.6 (Green’s current) .
• 04ZM Example Example 3.7 .
• 04ZN Proposition Proposition 3.8 .
• 04ZP Proof Proof.
• 04ZQ Notation Notation 3.9 .
• 04ZR Remark Remark 3.9.1 .
• 04ZS Theorem Theorem 3.10 .
• 04ZT Remark Remark 3.10.1 .
• 04ZU Remark Remark 3.10.2 .
• 04ZV Remark Remark 3.10.3 .
• 04ZW Remark Remark 3.10.4 .
• 04ZX Lemma Lemma 3.11 .
• 04ZY Proof Proof.
• 04ZZ Lemma Lemma 3.12 .
• 0500 Proof Proof.
• 0501 Lemma Lemma 3.13 .
• 0502 Proof Proof.
• 0503 Lemma Lemma 3.14 (Rearrangement Lemma) .
• 0504 Proof Proof.
• 0505 Proposition Proposition 3.15 .
• 0506 Proof Proof.
• 0507 Lemma Lemma 3.16 (Cancellation Lemma) .
• 0508 Proof Proof.
• 0509 Lemma Lemma 3.17 .
• 050A Proof Proof.
• 050B Lemma Lemma 3.18 .
• 050C Proof Proof.
• 050D Lemma Lemma 3.19 .
• 050E Proof Proof.
• 050F Proposition Proposition 3.20 .
• 050G Proof Proof.
• 050H Lemma Lemma 3.21 .
• 050I Proof Proof.
• 050J Lemma Lemma 3.22 .
• 050K Proof Proof.
• 050L Proof Proof of Theorem 3.10 .
• 050M Lemma Lemma 3.23 .
• 050N Proof Proof.
• 050P Subsection 3.3. Green’s currents on a cylinder
• 050Q Proposition Proposition 3.24 .
• 050R Proof Proof.
• 050S Corollary Corollary 3.24.1 .
• 050T Proof Proof.
• 050U Lemma Lemma 3.25 .
• 050V Proof Proof.
• 050W Proposition Proposition 3.26 .
• 050X Lemma Lemma 3.27 .
• 050Y Proof Proof.
• 050Z Proof Proof of Proposition 3.26 .
• 0510 Proposition Proposition 3.28 .
• 0511 Lemma Lemma 3.29 .
• 0512 Remark Remark 3.29.1 .
• 0513 Proof Proof of Lemma 3.29 .
• 0514 Proof Proof of Proposition 3.28 .
• 0515 Proposition Proposition 3.30 .
• 0516 Proof Proof.
• 0517 Subsection 3.4. A global existence result
• 0518 Proposition Proposition 3.31 .
• 0519 Remark Remark 3.31.1 .
• 051A Lemma Lemma 3.32 .
• 051B Proof Proof of Proposition 3.31 .
• 051C Lemma Lemma 3.33 .
• 051D Proof Proof.
• 051E Section 4. The approximately Calabi-Yau neck region
• 051F Subsection 4.1. Construction of a family of C 2 , α Kähler structures
• 051G Lemma Lemma 4.1 .
• 051H Proof Proof.
• 051I Lemma Lemma 4.2 .
• 051J Proof Proof.
• 051K Definition Definition 4.3 .
• 051L Remark Remark 4.3.1 .
• 051M Remark Remark 4.3.2 .
• 051N Lemma Lemma 4.4 .
• 051P Proof Proof.
• 051Q Proposition Proposition 4.5 .
• 051R Proof Proof.
• 051S Remark Remark 4.5.1 .
• 051T Lemma Lemma 4.6 .
• 051U Proof Proof.
• 051V Proposition Proposition 4.7 .
• 051W Proof Proof.
• 051X Subsection 4.2. Kähler geometry
• 051Y Subsubsection 4.2.1. The underlying complex manifold
• 051Z Proposition Proposition 4.8 .
• 0520 Remark Remark 4.8.1 .
• 0521 Proof Proof.
• 0522 Remark Remark 4.8.2 .
• 0523 Lemma Lemma 4.9 .
• 0524 Proof Proof.
• 0525 Lemma Lemma 4.10 .
• 0526 Proof Proof.
• 0527 Proposition Proposition 4.11 .
• 0528 Proof Proof.
• 0529 Corollary Corollary 4.11.1 .
• 052A Proof Proof.
• 052B Subsubsection 4.2.2. Kähler potentials
• 052C Proposition Proposition 4.12 .
• 052D Remark Remark 4.12.1 .
• 052E Proof Proof.
• 052F Remark Remark 4.12.2 .
• 052G Remark Remark 4.12.3 .
• 052H Lemma Lemma 4.13 .
• 052I Proof Proof.
• 052J Subsection 4.3. Geometries at regularity scales
• 052K Definition Definition 4.14 (Local regularity) .
• 052L Definition Definition 4.15 ( C k , α -regularity scale) .
• 052M Example Example 4.16 .
• 052N Example Example 4.17 .
• 052P Remark Remark 4.17.1 .
• 052Q Proposition Proposition 4.18 (Regularity scale on ℳ T ) .
• 052R Remark Remark 4.18.1 .
• 052S Remark Remark 4.18.2 .
• 052T Remark Remark 4.18.3 .
• 052U Corollary Corollary 4.18.1 (Harnack inequality for the regularity scale) .
• 052V Subsection 4.4. Fundamental estimates in the weighted Hölder spaces
• 052W Definition Definition 4.19 (Weight function) .
• 052X Remark Remark 4.19.1 .
• 052Y Remark Remark 4.19.2 .
• 052Z Remark Remark 4.19.3 .
• 0530 Lemma Lemma 4.20 (Lower bound estimate for the weight function) .
• 0531 Proof Proof.
• 0532 Definition Definition 4.21 (Weighted Hölder space) .
• 0533 Remark Remark 4.21.1 .
• 0534 Proposition Proposition 4.22 (Weighted Schauder estimate, the local version) .
• 0535 Remark Remark 4.22.1 .
• 0536 Proof Proof.
• 0537 Proposition Proposition 4.23 (Weighted error estimate) .
• 0538 Proof Proof.
• 0539 Subsection 4.5. Perturbation of complex structures
• 053A Proposition Proposition 4.24 .
• 053B Lemma Lemma 4.25 .
• 053C Proof Proof.
• 053D Remark Remark 4.25.1 .
• 053E Section 5. A Liouville Theorem on asymptotically Calabi spaces
• 053F Definition Definition 5.1 ( δ -aysmptotically Calabi space) .
• 053G Theorem Theorem 5.2 (Liouville Theorem) .
• 053H Remark Remark 5.2.1 .
• 053I Remark Remark 5.2.2 .
• 053J Subsection 5.1. Separation of variables and ODE reduction
• 053K Remark Remark 5.2.3 .
• 053L Remark Remark 5.2.4 .
• 053M Subsection 5.2. The case j k = 0 : uniform estimates and asymptotics
• 053N Proposition Proposition 5.3 .
• 053P Proof Proof.
• 053Q Corollary Corollary 5.3.1 .
• 053R Proof Proof.
• 053S Lemma Lemma 5.4 .
• 053T Proposition Proposition 5.5 .
• 053U Proof Proof.
• 053V Corollary Corollary 5.5.1 .
• 053W Subsection 5.3. The case j k ≠ 0 : uniform estimates and asymptotics
• 053X Proposition Proposition 5.6 .
• 053Y Proof Proof.
• 053Z Lemma Lemma 5.7 .
• 0540 Lemma Lemma 5.8 .
• 0541 Proof Proof.
• 0542 Proposition Proposition 5.9 .
• 0543 Proof Proof.
• 0544 Corollary Corollary 5.9.1 .
• 0545 Proposition Proposition 5.10 .
• 0546 Proof Proof.
• 0547 Lemma Lemma 5.11 .
• 0548 Proof Proof.
• 0549 Lemma Lemma 5.12 .
• 054A Remark Remark 5.12.1 .
• 054B Proof Proof.
• 054C Corollary Corollary 5.12.1 .
• 054D Subsection 5.4. Asymptotics of harmonic functions on the Calabi model space
• 054E Lemma Lemma 5.13 .
• 054F Proof Proof.
• 054G Proposition Proposition 5.14 (Asymptotics of harmonic functions) .
• 054H Proof Proof.
• 054I Subsection 5.5. The Poisson equation with prescribed asymptotics
• 054J Lemma Lemma 5.15 .
• 054K Proof Proof.
• 054L Proposition Proposition 5.16 .
• 054M Proof Proof.
• 054N Subsection 5.6. Proof of the Liouville theorem
• 054P Lemma Lemma 5.17 .
• 054Q Proof Proof of Theorem 5.2 .
• 054R Section 6. Perturbation to Calabi-Yau metrics on the neck
• 054S Subsection 6.1. Framework of perturbation
• 054T Lemma Lemma 6.1 (Implicit function theorem) .
• 054U Remark Remark 6.1.1 .
• 054V Lemma Lemma 6.2 .
• 054W Proof Proof.
• 054X Theorem Theorem 6.3 (Existence of S 1 -invariant Calabi-Yau metrics) .
• 054Y Lemma Lemma 6.4 (Nonlinear error estimate) .
• 054Z Proof Proof.
• 0550 Subsection 6.2. Some Liouville type theorems and removable singularity theorems
• 0551 Lemma Lemma 6.5 (Removable singularity) .
• 0552 Proof Proof.
• 0553 Lemma Lemma 6.6 (Liouville theorem on ℝ m + n ) .
• 0554 Proof Proof.
• 0555 Lemma Lemma 6.7 (Liouville theorem on a cylinder) .
• 0556 Subsection 6.3. Weighted analysis and existence of incomplete Calabi-Yau metrics
• 0557 Proposition Proposition 6.8 (Uniform injectivity estimate on the neck) .
• 0558 Proposition Proposition 6.9 (Weighted Schauder estimate on the neck, the global version) .
• 0559 Proof Proof.
• 055A Proposition Proposition 6.10 (Uniform injectivity estimate on the neck) .
• 055B Proof Proof.
• 055C Proof Proof of Theorem 6.3 .
• 055D Subsection 6.4. Geometric singularity and normalized limit measure
• 055E Definition Definition 6.11 (Measured Gromov-Hausdorff convergence) .
• 055F Remark Remark 6.11.1 .
• 055G Remark Remark 6.11.2 .
• 055H Remark Remark 6.11.3 .
• 055I Section 7. Proof of the main theorem
• 055J Subsection 7.1. Algebro-geometric aspect
• 055K Subsubsection 7.1.1. Poincaré residue
• 055L Subsubsection 7.1.2. A model partial resolution of singularities
• 055M Lemma Lemma 7.1 .
• 055N Proof Proof.
• 055P Subsubsection 7.1.3. A modification of the degenerating family
• 055Q Lemma Lemma 7.2 .
• 055R Proof Proof.
• 055S Lemma Lemma 7.3 .
• 055T Proof Proof.
• 055U Subsection 7.2. Tian-Yau metrics
• 055V Proposition Proposition 7.4 ( [ TY90 ] , see also [ HSVZ18 ] ) .
• 055W Lemma Lemma 7.5 .
• 055X Subsection 7.3. Construction of approximately Calabi-Yau metrics
• 055Y Remark Remark 7.5.1 .
• 055Z Subsubsection 7.3.1. Matching between the parameters t and T
• 0560 Subsubsection 7.3.2. Fixing the constants in the definition of weighted spaces
• 0561 Subsubsection 7.3.3. Construction of ω ⁡ ( t )
• 0562 Lemma Lemma 7.6 .
• 0563 Proof Proof.
• 0564 Proposition Proposition 7.7 .
• 0565 Proof Proof.
• 0566 Proposition Proposition 7.8 .
• 0567 Proof Proof.
• 0568 Proposition Proposition 7.9 .
• 0569 Proof Proof.
• 056A Proposition Proposition 7.10 .
• 056B Proposition Proposition 7.11 .
• 056C Proposition Proposition 7.12 .
• 056D Remark Remark 7.12.1 .
• 056E Subsection 7.4. Global weighted analysis on X ^ t and the proof of the main theorem
• 056F Proposition Proposition 7.13 (Nonlinear error estimate) .
• 056G Proposition Proposition 7.14 (Weighted Schauder estimate, the global version) .
• 056H Proposition Proposition 7.15 (Global injectivity estimates) .
• 056I Section 8. Extensions and Discussions
• 056J Subsection 8.1. More general situation
• 056K Lemma Lemma 8.1 .
• 056L Proof Proof.
• 056M Remark Remark 8.1.1 .
• 056N Subsection 8.2. Remarks and Questions
• 056P Section Appendix A Some formulae in special functions
• 056Q Subsection A.1. Modified Bessel functions
• 056R Lemma Lemma A.1 .
• 056S Proof Proof.
• 056T Subsection A.2. The confluent hypergeometric functions
• 056U Lemma Lemma A.2 .
• 056V Proof Proof.
• 056W Lemma Lemma A.3 .
• 056X Proof Proof.
• 056Y Lemma Lemma A.4 .
• 056Z Proof Proof.
• 0570 Lemma Lemma A.5 .
• 0571 Proof Proof.
• 0572 Lemma Lemma A.6 (Kummer’s transformation law) .
• 0573 Proof Proof.
• 0574 Lemma Lemma A.7 .
• 0575 Proof Proof.
• 0576 Lemma Lemma A.8 .
• 0577 Proof Proof.
• 0578 Corollary Corollary A.8.1 .
• 0579 Proof Proof.
• 057A Section 1 Introduction
• 057B Section 2 Statement of the main results
• 057C Theorem Theorem 1
• 057D Section 3 Proof of Theorem 1
• 057E Lemma Lemma 1
• 057F Section 1 Introduction
• 057G Theorem Theorem 1
• 057H Theorem Theorem 2
• 057I Section 2 Proof of Theorem 2
• 057J Lemma Lemma 1
• 057K Lemma Lemma 2
• 057L Section 3 Proof of Theorem 1
• 057M Theorem Theorem 3
• 057N Lemma Lemma 3
• 057P Section 4 Monge-Ampère equations
• 057Q Lemma Lemma 4
• 057R Lemma Lemma 5
• 057S Lemma Lemma 6
• 057T Theorem Theorem 4
• 057U Section 5 Fully non-linear and Hessian equations
• 057V Lemma Lemma 7
• 057W Lemma Lemma 8
• 057X Theorem Theorem 5
• 057Y Theorem Theorem 6
• 057Z Section 6 Trudinger Inequalities
• 0580 Theorem Theorem 7
• 0581 Section 1 Introduction
• 0582 Section 2 Chern-Simons-type functionals and critical points
• 0583 Section 3 Gauge equivalence and moment maps
• 0584 Subsection A symplectic example
• 0585 Lemma Lemma 3.4
• 0586 Lemma Lemma 3.6
• 0587 Subsection A holomorphic bundle example
• 0588 Section 4 Relationship to Kontsevich’s mirror conjecture
• 0589 Section 5 Stability
• 058A Definition Definition 5.1
• 058B Conjecture Conjecture 5.2
• 058C Section 6 The 2-torus
• 058D Section 1 Introduction
• 058E Section 2 Mean curvature flow
• 058F Lemma Lemma 2.1
• 058G Lemma Lemma 2.3
• 058H Section 3 Connect sums and Floer gradings
• 058I Subsection 3.1 The connect sum
• 058J Subsection 3.2 The grading on Floer cohomology
• 058K Section 4 Uniqueness
• 058L Lemma Lemma 4.2
• 058M Theorem Theorem 4.3
• 058N Section 5 Analogues of some properties of sheaves
• 058P Subsection 5.1 Twisting by line bundles
• 058Q Subsection 5.2 Stability of (S)Lags
• 058R Subsection 5.3 A Jordan-Hölder decomposition for Lagrangians
• 058S Definition Definition 5.3
• 058T Section 6 An example: families of affine quadrics
• 058U Lemma Lemma 6.6
• 058V Theorem Theorem 6.9
• 058W Section 7 The conjecture
• 058X Conjecture Conjecture 7.3
• 058Y Subsection Proof for our example
• 058Z Theorem Theorem 7.6
• 0590 Lemma Lemma 7.7
• 0591 Lemma Lemma 7.8
• 0592 Lemma Lemma 7.9
• 0593 Lemma Lemma 7.11
• 0594 Section 1. Introduction
• 0595 Theorem Theorem 1.1 .
• 0596 Theorem Theorem 1.2 .
• 0597 Proposition Proposition 1.3 .
• 0598 Section 2. Skeletons, formal models and divisors
• 0599 Definition Definition 2.1 .
• 059A Definition Definition 2.2 .
• 059B Definition Definition 2.3 .
• 059C Proposition Proposition 2.4 .
• 059D Proof Proof.
• 059E Proposition Proposition 2.5 .
• 059F Proof Proof.
• 059G SourceObject Construction 2.6 .
• 059H Remark Remark 2.7 .
• 059I Proposition Proposition 2.8 .
• 059J Proof Proof.
• 059K Corollary Corollary 2.9 .
• 059L Proof Proof.
• 059M Definition Definition 2.10 .
• 059N Proposition Proposition 2.11 .
• 059P Proof Proof.
• 059Q Remark Remark 2.12 .
• 059R Lemma Lemma 2.13 .
• 059S Proof Proof.
• 059T Section 3. Metrics
• 059U Definition Definition 3.1 .
• 059V Remark Remark 3.2 .
• 059W Proposition Proposition 3.3 .
• 059X Proof Proof.
• 059Y Definition Definition 3.4 .
• 059Z Proposition Proposition 3.5 .
• 05A0 Proof Proof.
• 05A1 Definition Definition 3.6 .
• 05A2 Proposition Proposition 3.7 .
• 05A3 Proof Proof.
• 05A4 Proposition Proposition 3.8 .
• 05A5 Proof Proof.
• 05A6 Definition Definition 3.9 .
• 05A7 Proposition Proposition 3.10 .
• 05A8 Proof Proof.
• 05A9 Proposition Proposition 3.11 .
• 05AA Proof Proof.
• 05AB Definition Definition 3.12 .
• 05AC Proposition Proposition 3.13 .
• 05AD Proof Proof.
• 05AE Section 4. Measures
• 05AF Definition Definition 4.1 .
• 05AG Definition Definition 4.2 .
• 05AH Remark Remark 4.4 .
• 05AI Proposition Proposition 4.5 .
• 05AJ Proof Proof.
• 05AK Lemma Lemma 4.6 .
• 05AL Proof Proof.
• 05AM Definition Definition 4.7 .
• 05AN Lemma Lemma 4.8 .
• 05AP Proof Proof.
• 05AQ Definition Definition 4.9 .
• 05AR Remark Remark 4.10 .
• 05AS Definition Definition 4.11 .
• 05AT Remark Remark 4.12 .
• 05AU Proposition Proposition 4.13 .
• 05AV Proof Proof.
• 05AW Remark Remark 4.14 .
• 05AX Corollary Corollary 4.15 .
• 05AY Proof Proof.
• 05AZ Remark Remark 4.16 .
• 05B0 Definition Definition 4.17 .
• 05B1 Remark Remark 4.18 .
• 05B2 Section 5. Comparison of the real and non-archimedean Monge-Ampère operator
• 05B3 Remark Remark 5.1 .
• 05B4 Theorem Theorem 5.2 .
• 05B5 Proof Proof.
• 05B6 Remark Remark 5.3 .
• 05B7 Remark Remark 5.4 .
• 05B8 Corollary Corollary 5.5 .
• 05B9 Proof Proof.
• 05BA Proposition Proposition 5.6 .
• 05BB Proof Proof.
• 05BC Corollary Corollary 5.7 .
• 05BD Proof Proof.
• 05BE Definition Definition 5.8 .
• 05BF Proposition Proposition 5.9 .
• 05BG Proof Proof.
• 05BH Corollary Corollary 5.10 .
• 05BI Proof Proof.
• 05BJ Section 6. Applications to regularity
• 05BK Definition Definition 6.1 .
• 05BL Proposition Proposition 6.2 .
• 05BM Proof Proof.
• 05BN Remark Remark 6.3 .
• 05BP Proposition Proposition 6.4 .
• 05BQ Proof Proof.
• 05BR Section Appendix A Reduction of germs
• 05BS Definition Definition A.1 .
• 05BT Definition Definition A.2 .
• 05BU Proposition Proposition A.3 .
• 05BV Proof Proof.
• 05BW Corollary Corollary A.4 .
• 05BX Proof Proof.
• 05BY Section Appendix B Convexity of psh-functions
• 05BZ Lemma Lemma B.1 .
• 05C0 Proof Proof.
• 05C1 Definition Definition B.2 .
• 05C2 Lemma Lemma B.3 .
• 05C3 Proof Proof.
• 05C4 Corollary Corollary B.4 .
• 05C5 Proof Proof.
• 05C6 Section 1. Introduction
• 05C7 Notation Notations .
• 05C8 Acknowledgement Acknowledgments .
• 05C9 Section 2. Ricci-flat metrics
• 05CA Subsection 2.1. Generalized Gibbons-Hawking ansatz
• 05CB Theorem Theorem 2.1 (cf. [ PP91 ] ) .
• 05CC Proof Proof.
• 05CD Subsection 2.2. Example: toric orbifold
• 05CE Subsection 2.3. Non-flat orbifold metrics
• 05CF Lemma Lemma 2.2 .
• 05CG Proof Proof.
• 05CH Subsection 2.4. Periodic solutions
• 05CI Definition Definition .
• 05CJ Proposition Proposition 2.3 .
• 05CK Proof Proof.
• 05CL Section 3. Limiting behavior of solutions
• 05CM Subsection 3.1. Exponential decay lemma
• 05CN Definition Definition .
• 05CP Definition Definition .
• 05CQ Conjecture Conjecture 3.1 (Exponential decay lemma) .
• 05CR Corollary Corollary 3.2 .
• 05CS Subsection 3.2. The semi-flat case
• 05CT Subsection 3.3. Two dimensional example: local K3 (after [ OV96 ] and [ GW00 ] )
• 05CU Subsection 3.4. Higher dimensional case
• 05CV Section 4. Local mirror symmetry and Legendre transform
• 05CW Subsection 4.1. Linear algebra of Legendre transform and Monge-Ampére equations
• 05CX Lemma Lemma 4.1 .
• 05CY Proof Proof.
• 05CZ Subsection 4.2. Monge-Ampère manifolds
• 05D0 Definition Definition .
• 05D1 Theorem Theorem 4.2 .
• 05D2 Proof Proof.
• 05D3 Remark Remark .
• 05D4 Section 1. Introduction
• 05D5 Theorem Theorem 1.1 .
• 05D6 Remark Remark 1.2 .
• 05D7 Remark Remark 1.3 .
• 05D8 Remark Remark 1.4 .
• 05D9 Theorem Theorem 1.5 .
• 05DA Section 2. Preliminaries
• 05DB Subsection 2.1. Cheeger-Gromov convergence
• 05DC Theorem Theorem 2.1 (Kähler version of Cheeger-Gromov convergence theorem) .
• 05DD Conjecture Conjecture 2.2 .
• 05DE Subsection 2.2. Implicit function theorem
• 05DF Theorem Theorem 2.3 (Theorem 3.2 in [ 31 ] ) .
• 05DG Section 3. The blow-up limit
• 05DH Lemma Lemma 3.1 .
• 05DI Proof Proof.
• 05DJ Lemma Lemma 3.2 .
• 05DK Proof Proof.
• 05DL Lemma Lemma 3.3 .
• 05DM Proof Proof.
• 05DN Proposition Proposition 3.4 .
• 05DP Proof Proof.
• 05DQ Remark Remark 3.5 .
• 05DR Remark Remark 3.6 .
• 05DS Section 4. Local special lagrangian fibrations
• 05DT SourceObject Condition 4.1 .
• 05DU Lemma Lemma 4.2 .
• 05DV Proof Proof.
• 05DW Lemma Lemma 4.3 .
• 05DX Proof Proof.
• 05DY Lemma Lemma 4.4 .
• 05DZ Proof Proof.
• 05E0 Lemma Lemma 4.5 .
• 05E1 Proof Proof.
• 05E2 Proposition Proposition 4.6 .
• 05E3 Proof Proof.
• 05E4 Section 5. Proof of Theorem 1.1
• 05E5 Proof Proof of Theorem 1.1 .
• 05E6 Section 6. Estimates for injectivity radius
• 05E7 Theorem Theorem 6.1 (Theorem 2.0.1. in [ 16 ] ) .
• 05E8 Corollary Corollary 6.2 .
• 05E9 Proof Proof.
• 05EA Corollary Corollary 6.3 .
• 05EB Chapter The monomial-divisor mirror map
• 05EC Bibliography Paul S. Aspinwall, Brian R. Greene, and David R. Morrison. The monomial-divisor mirror map. Internat. Math. Res. Notices, (12):319–337, 1993.
• 05ED Chapter Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties
• 05EE Bibliography Victor V. Batyrev. Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Alg. Geom., 3:493–535, 1994.
• 05EF Chapter Topological methods
• 05EG Bibliography Anders Bjorner. Topological methods. In R. L. Graham, editor, Handbook of Combinatorics, pages 1819–1872. Elsevier, 1995.
• 05EH Chapter The geometry of toric varieties
• 05EI Bibliography Vladimir I. Danilov. The geometry of toric varieties. Uspekhi Mat. Nauk, 33(2(200)):85–134, 247, 1978.
• 05EJ Bibliography William Fulton. Introduction to Toric Varieties, volume 131 of Annals of Math. Studies. Princeton University Press, 1993.
• 05EK Bibliography Israel M. Gelfand, Mikhail M. Kapranov, and Andrei V. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhauser, 1994.
• 05EL Chapter Cell decomposition of polytopes by bending
• 05EM Bibliography Jacob E. Goodman and Janos Pach. Cell decomposition of polytopes by bending. Israel J. Math., 64(2):129–138, 1988.
• 05EN Bibliography Mark Gross. Topological mirror symmetry. Invent. Math., 144(1):75–137, 2001.
• 05EP Chapter Convex Polytopes
• 05EQ Bibliography Branko Grunbaum. Convex Polytopes. Interscience, London, 1967.
• 05ER Chapter Kahler structures on toric varieties
• 05ES Bibliography Victor Guillemin. Kahler structures on toric varieties. J. Differential Geom., 40(2):285–309, 1994.
• 05ET Bibliography Mark Gross and Pelham M.H. Wilson. Large complex structure limits of K3 surfaces. J. Differential Geom., 55(3):475–546, 2000.
• 05EU Chapter Amoebas over non-archimedean fields
• 05EV Bibliography Mikhail Kapranov. Amoebas over non-archimedean fields. Preprint, 2000.
• 05EW Chapter Lectures at Rutgers University
• 05EX Bibliography Maxim Kontsevich. Lectures at Rutgers University. October 2000.
• 05EY Bibliography Maxim Kontsevich and Yan Soibelman. Homological mirror symmetry and torus fibrations. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 203–263. World Sci. Publishing, River Edge, NJ, 2001.
• 05EZ Chapter Subdivisions and triangulations of polytopes
• 05F0 Bibliography Carl W. Lee. Subdivisions and triangulations of polytopes. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, pages 271–290. CRC–Press, New York, 1997.
• 05F1 Bibliography Grigory Mikhalkin. Amoebas of algebraic varieties. Preprint math.AG/0108225, 2001.
• 05F2 Bibliography Grigory Mikhalkin. Decomposition into pairs-of-pants for complex algebraic hypersurfaces. Preprint math.GT/0205011, 2002.
• 05F3 Chapter Lectures at RIMS Kyoto
• 05F4 Bibliography David R. Morrison. Lectures at RIMS Kyoto. June 2000.
• 05F5 Chapter Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties
• 05F6 Bibliography Tadao Oda. Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer–Verlag, 1988.
• 05F7 Chapter Lagrangian torus fibrations of toric Calabi-Yau manifolds I
• 05F8 Bibliography Wei-Dong Ruan. Lagrangian torus fibrations of toric Calabi-Yau manifolds I. Preprint math.DG/9904012, 1999.
• 05F9 Chapter Lagrangian torus fibrations and mirror symmetry of Calabi-Yau hypersurfaces in toric varieties
• 05FA Bibliography Wei-Dong Ruan. Lagrangian torus fibrations and mirror symmetry of Calabi-Yau hypersurfaces in toric varieties. Preprint math.DG/0007028, 2000.
• 05FB Chapter Decomposability of polytopes and polyhedra
• 05FC Bibliography Zeev Smilansky. Decomposability of polytopes and polyhedra. Geom. Dedicata, 24(1):29–49, 1987.
• 05FD Bibliography Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Mirror symmetry is T-duality. Nuclear Phys. B, 479(1-2):243–259, 1996.
• 05FE Chapter Three-dimensional geometry and topology
• 05FF Bibliography William P. Thurston. Three-dimensional geometry and topology. Lecture notes, available at http://msri.org/publications/books/gt3m/, 1980.
• 05FG Chapter Torus fibrations of Calabi-Yau hypersurfaces in toric varieties
• 05FH Bibliography Ilia Zharkov. Torus fibrations of Calabi-Yau hypersurfaces in toric varieties. Duke Math. J., 101(2):237–257, 2000.
• 05FI Bibliography William Fulton. Introduction to Toric Varieties, volume 131 of Annals of Math. Studies. Princeton University Press, 1993.
• 05FJ Chapter The radiance obstruction and parallel forms on affine manifolds
• 05FK Bibliography William Goldman and Morris W. Hirsch. The radiance obstruction and parallel forms on affine manifolds. Trans. Amer. Math. Soc., 286(2):629–649, 1984.
• 05FL Bibliography Israel M. Gelfand, Mikhail M. Kapranov, and Andrei V. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhauser, 1994.
• 05FM Chapter Affine manifolds and mirror symmetry
• 05FN Bibliography Mark Gross and Bernd Siebert. Affine manifolds and mirror symmetry. In preparation, 2002.
• 05FP Bibliography Christian Haase and Ilia Zharkov. Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I. Preprint math.AG/0205321, 2002.
• 05FQ Bibliography Maxim Kontsevich and Yan Soibelman. Homological mirror symmetry and torus fibrations. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 203–263. World Sci. Publishing, River Edge, NJ, 2001.
• 05FR Bibliography Maxim Kontsevich and Yury Tschinkel. Non-Archimedean Kahler geometry. In preparation, 2002.
• 05FS Bibliography Ilia Zharkov. Limiting behavior of local Calabi-Yau metrics. In preparation, 2002.
• 05FT Chapter C^\alpha-compactness for manifolds with Ricci curvature and injectivity radius bounded below
• 05FU Bibliography Michael T. Anderson and Jeff Cheeger, C^\alpha-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 (1992), no. 2, 265–281.
• 05FV Bibliography Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445.
• 05FW Chapter On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth
• 05FX Bibliography Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313–349.
• 05FY Chapter Lower bounds on Ricci curvature and the almost rigidity of warped products
• 05FZ Bibliography Jeff Cheeger and Tobias H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189–237.
• 05G0 Chapter On the structure of spaces with Ricci curvature bounded below. I
• 05G1 Bibliography , On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480.
• 05G2 Chapter Gravitational instantons with faster than quadratic curvature decay (II)
• 05G3 Bibliography Gao Chen and Xiuxiong Chen, Gravitational instantons with faster than quadratic curvature decay (II), arXiv.org:1508.07908, 2015.
• 05G4 Chapter Gravitational instantons with faster than quadratic curvature decay (III)
• 05G5 Bibliography , Gravitational instantons with faster than quadratic curvature decay (III), arXiv.org:1603.08465, 2016.
• 05G6 Chapter On the singularities of spaces with bounded Ricci curvature
• 05G7 Bibliography Jeff Cheeger, Tobias H. Colding, and Gang Tian, On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal. 12 (2002), no. 5, 873–914.
• 05G8 Chapter Volume estimates for Kahler-Einstein metrics and rigidity of complex structures
• 05G9 Bibliography Xiuxiong Chen and Simon K. Donaldson, Volume estimates for Kahler-Einstein metrics and rigidity of complex structures, J. Differential Geom. 93 (2013), no. 2, 191–201.
• 05GA Bibliography Jeff Cheeger, Kenji Fukaya, and Mikhael Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 327–372.
• 05GB Bibliography Jeff Cheeger and Mikhael Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986), no. 3, 309–346.
• 05GC Bibliography , Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990), no. 1, 269–298.
• 05GD Chapter The splitting theorem for manifolds of nonnegative Ricci curvature
• 05GE Bibliography Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128.
• 05GF Chapter Asymptotically conical Calabi-Yau manifolds, I
• 05GG Bibliography Ronan J. Conlon and Hans-Joachim Hein, Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J. 162 (2013), 2855–2902.
• 05GH Chapter Asymptotically conical Calabi-Yau metrics on quasi-projective varieties
• 05GI Bibliography , Asymptotically conical Calabi-Yau metrics on quasi-projective varieties, GAFA 25 (2015), 517–552.
• 05GJ Chapter Integral bounds on curvature elliptic estimates and rectifiability of singular sets
• 05GK Bibliography Jeff Cheeger, Integral bounds on curvature elliptic estimates and rectifiability of singular sets, Geom. Funct. Anal. 13 (2003), no. 1, 20–72.
• 05GL Chapter Lower bounds on Ricci curvature and quantitative behavior of singular sets
• 05GM Bibliography Jeff Cheeger and Aaron Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math. 191 (2013), no. 2, 321–339.
• 05GN Bibliography , Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. (2) 182 (2015), no. 3, 1093–1165.
• 05GP Bibliography Jeff Cheeger and Gang Tian, Anti-self-duality of curvature and degeneration of metrics with special holonomy, Comm. Math. Phys. 255 (2005), no. 2, 391–417.
• 05GQ Bibliography , Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 487–525.
• 05GR Chapter Differential equations on Riemannian manifolds and their geometric applications
• 05GS Bibliography Shiu-Yuen Cheng and Shing-Tung Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354.
• 05GT Chapter Two-forms on four-manifolds and elliptic equations
• 05GU Bibliography Simon K. Donaldson, Two-forms on four-manifolds and elliptic equations, Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 153–172.
• 05GV Chapter Calabi-Yau metrics on Kummer surfaces as a model gluing problem
• 05GW Bibliography , Calabi-Yau metrics on Kummer surfaces as a model gluing problem, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 109–118.
• 05GX Chapter The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups
• 05GY Bibliography Christopher Deninger and Wilhelm Singhof, The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups, Invent. Math. 78 (1984), no. 1, 101–112.
• 05GZ Chapter Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds
• 05H0 Bibliography Lorenzo Foscolo, Mark Haskins, and Johannes Nordstrom, Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds, arXiv.org: 1709.04904, 2017.
• 05H1 Chapter Self-dual conformal structures on l C P^2
• 05H2 Bibliography Andreas Floer, Self-dual conformal structures on l C P^2, J. Differential Geom. 33 (1991), no. 2, 551–573.
• 05H3 Chapter The space of hyperkahler metrics on a 4-manifold with boundary
• 05H4 Bibliography Joel Fine, Jason D. Lotay, and Michael Singer, The space of hyperkahler metrics on a 4-manifold with boundary, Forum Math. Sigma 5 (2017), e6, 50.
• 05H5 Bibliography Lorenzo Foscolo, ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface, arXiv.org: 1603.06315, 2016.
• 05H6 Chapter Collapsing Riemannian manifolds to ones of lower dimensions
• 05H7 Bibliography Kenji Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987), no. 1, 139–156.
• 05H8 Chapter A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters
• 05H9 Bibliography , A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988), no. 1, 1–21.
• 05HA Chapter The fundamental groups of almost non-negatively curved manifolds
• 05HB Bibliography Kenji Fukaya and Takao Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2) 136 (1992), no. 2, 253–333.
• 05HC Chapter Principles of algebraic geometry
• 05HD Bibliography Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994, Reprint of the 1978 original.
• 05HE Chapter Stringy cosmic strings and noncompact Calabi-Yau manifolds
• 05HF Bibliography Brian R. Greene, Alfred Shapere, Cumrun Vafa, and Shing-Tung Yau, Stringy cosmic strings and noncompact Calabi-Yau manifolds, Nuclear Phys. B 337 (1990), no. 1, 1–36.
• 05HG Chapter The spectrum of the Laplacian on Riemannian Heisenberg manifolds
• 05HH Bibliography Carolyn S. Gordon and Edward N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), no. 2, 253–271.
• 05HI Bibliography Mark Gross and P. M. H. Wilson, Large complex structure limits of K3 surfaces, J. Differential Geom. 55 (2000), no. 3, 475–546.
• 05HJ Chapter Gravitational instantons from rational elliptic surfaces
• 05HK Bibliography Hans-Joachim Hein, Gravitational instantons from rational elliptic surfaces, J. Amer. Math. Soc. 25 (2012), no. 2, 355–393.
• 05HL Chapter Type II degeneration of Ricci-flat metrics on K3 surfaces
• 05HM Bibliography Hans-Joachim Hein, Song Sun, Jeff Viaclovsky, and Ruobing Zhang, Type II degeneration of Ricci-flat metrics on K3 surfaces, in preparation.
• 05HN Chapter Massive string theories from M-theory and F-theory
• 05HP Bibliography Chris M. Hull, Massive string theories from M-theory and F-theory, J. High Energy Phys. (1998), no. 11, Paper 27, 10.
• 05HQ Chapter On asymptotics of complete Ricci-flat Kahler metrics on open manifolds
• 05HR Bibliography Bert Koehler and Marco Kuhnel, On asymptotics of complete Ricci-flat Kahler metrics on open manifolds, Manuscripta Math. 132 (2010), no. 3-4, 431–462.
• 05HS Chapter Ricci-flat Kahler metrics on affine algebraic manifolds and degenerations of Kahler-Einstein K3 surfaces
• 05HT Bibliography Ryoichi Kobayashi, Ricci-flat Kahler metrics on affine algebraic manifolds and degenerations of Kahler-Einstein K3 surfaces, Kahler metric and moduli spaces, Adv. Stud. Pure Math., vol. 18, Academic Press, Boston, MA, 1990, pp. 137–228. \MR1145249
• 05HU Chapter Twisted connected sums and special Riemannian holonomy
• 05HV Bibliography Alexei Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003), 125–160. \MR2024648
• 05HW Chapter The construction of ALE spaces as hyper-Kahler quotients
• 05HX Bibliography Peter B. Kronheimer, The construction of ALE spaces as hyper-Kahler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683.
• 05HY Chapter Gluing theorems for complete anti-self-dual spaces
• 05HZ Bibliography Alexei Kovalev and Michael Singer, Gluing theorems for complete anti-self-dual spaces, Geom. Funct. Anal. 11 (2001), no. 6, 1229–1281.
• 05I0 Chapter Special functions and their applications
• 05I1 Bibliography N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972, Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.
• 05I2 Bibliography Claude LeBrun, Complete Ricci-flat Kahler metrics on C^n need not be flat, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 297–304. \MR1128554
• 05I3 Chapter The collapsing geometry of almost Ricci-flat 4-manifolds
• 05I4 Bibliography John Lott, The collapsing geometry of almost Ricci-flat 4-manifolds, arXiv.org: 1708.06780, 2017.
• 05I5 Chapter A Kummer-type construction of self-dual 4-manifolds
• 05I6 Bibliography Claude LeBrun and Michael Singer, A Kummer-type construction of self-dual 4-manifolds, Math. Ann. 300 (1994), no. 1, 165–180.
• 05I7 Chapter Symmetric Green's functions on complete manifolds
• 05I8 Bibliography Peter Li and Luen-Fai Tam, Symmetric Green's functions on complete manifolds, Amer. J. Math. 109 (1987), no. 6, 1129–1154.
• 05I9 Chapter Rigidity for multi-Taub-NUT metrics
• 05IA Bibliography Vincent Minerbe, Rigidity for multi-Taub-NUT metrics, J. Reine Angew. Math. 656 (2011), 47–58.
• 05IB Chapter Empty-space generalization of the Schwarzschild metric
• 05IC Bibliography Erza T. Newman, Louis A. Tamburino, and Theodore W.J. Unti, Empty-space generalization of the Schwarzschild metric, J. Mathematical Phys. 4 (1963), 915–923.
• 05ID Chapter Topology and \varepsilon-regularity theorems on collapsed manifolds with Ricci curvature bounds
• 05IE Bibliography Aaron Naber and Ruobing Zhang, Topology and \varepsilon-regularity theorems on collapsed manifolds with Ricci curvature bounds, Geom. Topol. 20 (2016), no. 5, 2575–2664.
• 05IF Bibliography Hirosi Ooguri and Cumrun Vafa, Summing up Dirichlet instantons, Phys. Rev. Lett. 77 (1996), no. 16, 3296–3298.
• 05IG Chapter Collapsed manifolds with bounded sectional curvature and applications
• 05IH Bibliography Xiaochun Rong, Collapsed manifolds with bounded sectional curvature and applications, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 1–23.
• 05II Chapter Non-minimal scalar-flat Kahler surfaces and parabolic stability
• 05IJ Bibliography Yann Rollin and Michael Singer, Non-minimal scalar-flat Kahler surfaces and parabolic stability, Invent. Math. 162 (2005), no. 2, 235–270.
• 05IK Chapter The geometries of 3-manifolds
• 05IL Bibliography Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487.
• 05IM Chapter A comprehensive introduction to differential geometry. Vol. I
• 05IN Bibliography Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, second ed., Publish or Perish, Inc., Wilmington, Del., 1979.
• 05IP Chapter Projective embedding of log riemann surfaces and k-stability
• 05IQ Bibliography Jingzhou Sun and Song Sun, Projective embedding of log riemann surfaces and k-stability, arXiv.org: 1605.01089, 2016.
• 05IR Chapter Empty space-times admitting a three parameter group of motions
• 05IS Bibliography Abraham H. Taub, Empty space-times admitting a three parameter group of motions, Gen. Relativity Gravitation 36 (2004), no. 12, 2699–2719, Reprint of Ann. of Math. (2) 53 (1951), 472–490.
• 05IT Chapter On Calabi's conjecture for complex surfaces with positive first Chern class
• 05IU Bibliography Gang Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172.
• 05IV Chapter Complete Kahler manifolds with zero Ricci curvature. I
• 05IW Bibliography Gang Tian and Shing-Tung Yau, Complete Kahler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc. 3 (1990), no. 3, 579–609.
• 05IX Bibliography Shing-Tung Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.
• 05IY Chapter Second order families of special Lagrangian 3-folds
• 05IZ Bibliography R.L. Bryant, Second order families of special Lagrangian 3-folds, math.DG/0007128, 2000.
• 05J0 Bibliography E. Goldstein, Calibrated fibrations, math.DG/9911093, 1999.
• 05J1 Chapter Calibrated fibrations on complete manifolds via torus action
• 05J2 Bibliography E. Goldstein, Calibrated fibrations on complete manifolds via torus action, math.DG/0002097, 2000.
• 05J3 Chapter Special Lagrangian submanifolds and algebraic complexity one torus actions
• 05J4 Bibliography E. Goldstein, Special Lagrangian submanifolds and algebraic complexity one torus actions, math.DG/0003220, 2000.
• 05J5 Chapter Special Lagrangian fibrations I: Topology
• 05J6 Bibliography M. Gross, Special Lagrangian fibrations I: Topology. In M.-H. Saito, Y. Shimizu, and K. Ueno, editors, Integrable Systems and Algebraic Geometry, pages 156–193, Singapore, 1998. World Scientific. alg-geom/9710006.
• 05J7 Bibliography M. Gross, Special Lagrangian fibrations II: Geometry, math.AG/9809072, 1998.
• 05J8 Bibliography M. Gross, Topological mirror symmetry, math.AG/9909015, 1999.
• 05J9 Chapter Mirror symmetry via 3-tori for a class of Calabi–Yau threefolds
• 05JA Bibliography M. Gross and P.M.H. Wilson, Mirror symmetry via 3-tori for a class of Calabi–Yau threefolds, Math. Ann. 309 (1997), 505–531. alg-geom/9608004.
• 05JB Bibliography R. Harvey and H.B. Lawson, Calibrated geometries, Acta Mathematica 148 (1982), 47–157.
• 05JC Chapter On counting special Lagrangian homology 3-spheres
• 05JD Bibliography D.D. Joyce, On counting special Lagrangian homology 3-spheres, hep-th/9907013, 1999.
• 05JE Chapter Special Lagrangian m-folds in \mathbb C^m with symmetries
• 05JF Bibliography D.D. Joyce, Special Lagrangian m-folds in \mathbb C^m with symmetries, math.DG/0008021, 2000.
• 05JG Chapter Constructing special Lagrangian m-folds in \mathbb C^m by evolving quadrics
• 05JH Bibliography D.D. Joyce, Constructing special Lagrangian m-folds in \mathbb C^m by evolving quadrics, math.DG/0008155, 2000.
• 05JI Chapter Evolution equations for special Lagrangian 3-folds in \mathbb C^3
• 05JJ Bibliography D.D. Joyce, Evolution equations for special Lagrangian 3-folds in \mathbb C^3, math.DG/0010036, 2000.
• 05JK Bibliography R.C. McLean, Deformations of calibrated submanifolds, Communications in Analysis and Geometry 6 (1998), 705–747.
• 05JL Chapter The geometry underlying mirror symmetry
• 05JM Bibliography D.R. Morrison, The geometry underlying mirror symmetry, pages 283–310 in New trends in algebraic geometry, editors K. Hulek, F. Catenese, C. Peters and M. Reid, L.M.S. Lecture Notes Series 264, C.U.P., 1999. alg-geom/9608006.
• 05JN Chapter Compactifications of moduli spaces inspired by mirror symmetry
• 05JP Bibliography D.R. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Asterisque 218 (1993), 243–271. alg-geom/9304007.
• 05JQ Chapter Lectures on nonlinear evolution equations
• 05JR Bibliography R. Racke, Lectures on nonlinear evolution equations, Aspects of Math. E19, Max-Planck Institute, Bonn, 1992.
• 05JS Bibliography W.-D. Ruan, Lagrangian tori fibration of toric Calabi–Yau manifold I, math.DG/9904012, 1999.
• 05JT Chapter Lagrangian tori fibration of toric Calabi–Yau manifold III: symplectic topological SYZ mirror construction for general quintics
• 05JU Bibliography W.-D. Ruan, Lagrangian tori fibration of toric Calabi–Yau manifold III: symplectic topological SYZ mirror construction for general quintics, math.DG/9909126, 1999.
• 05JV Bibliography W.-D. Ruan, Lagrangian torus fibration and mirror symmetry of Calabi–Yau hypersurface in toric variety, math.DG/0007028, 2000.
• 05JW Chapter Lagrangian torus fibration of quintic Calabi–Yau hypersurfaces II: technical results on gradient flow construction
• 05JX Bibliography W.-D. Ruan, Lagrangian torus fibration of quintic Calabi–Yau hypersurfaces II: technical results on gradient flow construction, preprint, 2000.
• 05JY Bibliography A. Strominger, S.-T. Yau, and E. Zaslow, Mirror symmetry is T-duality, Nuclear Physics B479 (1996), 243–259. hep-th/9606040.
• 05JZ Bibliography I. Zharkov, Torus fibrations of Calabi–Yau hypersurfaces in toric varieties and mirror symmetry, Duke Math. J.\ 101 (2000), 237–257. alg-geom/9806091.