An $n$-dimensional Calabi-Yau (CY) manifold $(X,\omega,\Omega)$ is a Kähler manifold with a nowhere vanishing holomorphic volume form $\Omega$, satisfying the complex Monge-Ampère (MA) equation

which implies the Ricci flatness of the metric. Furthermore, such manifolds admit parallel spinors, hence are candidates for the target space metrics of supersymmetric type II string theories. Special Lagrangians of phase $\theta$ are $n$-dimensional submanifolds $L$ satisfying

These are absolute minimizers within their homology classes, due to the calibration inequality

saturated precisely by the special Lagrangians. Physically, these correspond to the support of BPS D-branes.

The Strominger-Yau-Zaslow conjecture [67] in its primitive form asks:

The physical origin of the SYZ conjecture [67] comes largely from mirror symmetry, and a very brief sketch is as follows. From homological mirror symmetry, one expects a compact Calabi-Yau manifold $X$ admits a mirror $X^{\vee}$, such that the category of D-branes on both sides are identified. On the holomorphic side $X^{\vee}$ (‘B-side’), the points $x\in X^{\vee}$ support skyscrapper sheaves $\mathcal{O}_{x}$, which should correspond to certain Lagrangian branes inside the symplectic side $X$ (‘A-side’). The extension groups $\text{Ext}^{*}(\mathcal{O}_{x},\mathcal{O}_{x})\simeq H^{*}(T^{n})$, which suggests the Lagrangian branes are torus objects. For $x\neq y$, the Ext groups between $\mathcal{O}_{x},\mathcal{O}_{y}$ would vanish, which suggests (inconclusively¹¹ 1 The vanishing of Floer cohomology does not imply the vanishing of Floer cochain spaces, and there seems to be no strong argument to rule out intersecting special Lagrangians.) that the tori are disjoint, leading to the speculation of the Lagrangian fibration structure. The assertion about special Lagrangians, is however beyond mere homological mirror symmetry, and comes from the BPS condition on the D-branes. The moduli space of all $\mathcal{O}_{x}$ is the mirror manifold $X^{\vee}$, which should then be identified with the moduli space of BPS branes supported on the special Lagrangian tori. This moduli interpretation gives rise to a zeroth order approximation of Kähler structure on the mirror manifold $X^{\vee}$, subject to the higher order corrections related to the holomorphic discs (‘instanton corrections’), whose effect is supposedly exponentially suppressed except near the singular fibres. Ignoring the subtleties of singular fibres, then the SYZ picture offers a program to reconstruct the mirror, and interpret homological mirror symmetry as a version of Fourier-Mukai transform (‘Mirror symmetry is T-duality’).

Differential geometrically, the main evidence presented in the SYZ paper is the semiflat metrics. Consider the logarithm map $\text{Log}_{t}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$ over some open convex subset $U\subset\mathbb{R}^{n}$,

Imposing $T^{n}$ symmetry, then by an elementary Hessian computation, $\phi$ is a smooth strictly convex function downstairs on $U$ if and only if its pullback to $\text{Log}_{t}^{-1}(U)$ is a smooth Kähler potential, and the Calabi-Yau condition

is equivalent to the real Monge-Ampère equation

In this setting, the metric $dd^{c}\phi\circ\text{Log}_{t}$ is called semiflat, because its restriction to the $T^{n}$ fibres are Euclidean, due to $T^{n}$-symmetry. With respect to the Calabi-Yau structure

the $T^{n}$ fibres are special Lagrangians of phase zero. The purpose of introducing the normalization parameter $t$, is that as $t\to 0$, the $T^{n}$-fibres shrink down to zero size, and the Calabi-Yau metrics $dd^{c}(\phi\circ\text{Log}_{t})$ converge to the real Monge-Ampère metric on $U\subset\mathbb{R}^{n}$

Now near the large complex structure limit, which is a certain limiting situation for a family of Calabi-Yau metrics, it is expected that the semiflat metrics emerge as an asymptotic description of the degenerating Calabi-Yau metrics, in the generic region of the Calabi-Yau manifolds. In the SYZ picture, the generic region heuristically means away from the singular special Lagrangian fibres. The main point is that in the generic region, the special Lagrangians are just small perturbations of the logarithm maps in local toric charts. As we approach the large complex structure limit, the percentage of the Calabi-Yau volume measure occupied by the generic region should tend to $100\%$.

This recount of this SYZ heuristic reasoning underlines a few precautions:

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  The metric predictions are more compelling in the generic region. The non-generic region is subject to instanton correction effects, whose metric significance is much more difficult to analyze.

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  The SYZ fibration is likely an emergent behaviour near the large complex structure limit. In particular, the conjecture is concerned with a family of Calabi-Yau manifolds, and features such as semiflat metrics would only appear sufficiently close to the limit.

Aside from its importance in mirror symmetry, the SYZ conjecture is interesting for other diverse fields such as minimal surface theory, Riemannian geometry, Kähler and algebraic geometry.

• •

  Special Lagrangians are minimal submanifolds, and in fact calibrated submanifolds. Currently, there are few methods for producing special Lagrangians in sufficiently large supply of Calabi-Yau manifolds, although there is a series of conjectures initiated by Thomas-Yau [75][74] and further developed in [43][57].

• •

  The behaviour of a family of Einstein metrics depends strongly on whether the volume of geodesic balls satisfies the noncollapsing condition for some uniform constant $\kappa>0$.

  $$\text{Vol}_{g}(B_{g}(r))\geq\kappa r^{\dim_{\mathbb{R}}X},\quad\forall 0<r\leq\text{diam}(X).$$

  A good convergence and regularity theory is available in the non-collapsing case [13]. On the other hand, metric degeneration in the collapsing case is largely terra incognita in Riemannian geometry, and the semiflat metric asymptote is a highly nontrivial emergent feature for a collapsing family of Calabi-Yau metrics.

• •

  A recurring theme of Kähler geometry is the interplay between metric and complex geometry. Käher-Einstein metrics in the non-collapsing case is tied to projective geometry [23]. The large complex structure limit is a very severe kind of polarized degeneration, whose transcendental behaviour (related to exponential and logarithms) is not adequately captured by traditional projective geometry, and instead non-archimedean geometry stands out as a natural framework. One can then ask about the relation between Calabi-Yau metrics and non-archimedean geometry, a problem that turns out to be related to the SYZ conjecture.

An influential early work related to the SYZ conjecture is the gluing description of Gross-Wilson [32] concerning the hyperkähler metric on K3 surfaces $X$ with elliptic fibrations $\pi:X\to\mathbb{CP}^{1}$.

By hyperkähler rotation, the special Lagrangian torus fibres can be viewed as the holomorphic elliptic curves in a different complex structure. In the generic situation, the elliptic fibration has 24 $I_{1}$-type singular fibres, namely the local singularity in the fibration is modelled on $(z_{1},z_{2})\mapsto z_{1}z_{2}$ complex geometrically. They fix Kähler classes $[\omega_{X}]$ on the K3 surface, and $[\omega_{\mathbb{P}^{1}}]$ on $\mathbb{CP}^{1}$, and describe the Calabi-Yau metrics $\omega_{\tau}$ in the Kähler class $\tau[\omega_{X}]+\pi^{*}[\omega_{\mathbb{P}^{1}}]$ for $0<\tau\ll 1$. Some key conceptual features are:

• •

  In the generic region, the metrics $\omega_{\tau}$ are up to exponentially small errors modelled on semiflat metrics, which can be explicitly described via the periods integrals of the elliptic curves. The subset of the K3 surface on which the semiflat metric asymptote breaks down, has length scale $O(\tau^{1/2}|\log\tau|^{1/2})$.

• •

  As $\tau\to 0$, the metrics $\omega_{\tau}$ converge in the Gromov-Hausdorff sense to a singular metric on $\mathbb{CP}^{1}$. The diameter of the K3 surfaces is of constant order $O(1)$. The length scale of generic elliptic curve fibres is $O(\tau^{1/2})$, which shrinks to zero size as $\tau\to 0$.

• •

  In the neighbourhood of the $I_{1}$ type singular fibres, the metrics are modelled on the Ooguri-Vafa metrics, which are explicit $S^{1}$-invariant incomplete hyperkähler metrics constructed by means of the Gibbons-Hawking ansatz.

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  The asymptotic geometry of the Ooguri-Vafa metric matches with the semiflat metric. This is a basic requirement for the gluing construction.

• •

  Near the nodal singular point of the $I_{1}$-fibre, the Ooguri-Vafa metric contains a small region approximated by the Taub-NUT metric, whose length scale is $O(\tau^{1/2}|\log\tau|^{-1/2})$. These regions concentrate almost the entire $L^{2}$-Riemannian curvature of the K3 surface.

The Gross-Wilson picture represents the best hope on the SYZ conjecture,²² 2 As a caveat sometimes overlooked in the literature, the hyperkähler rotation of the Gross-Wilson setting is not quite a polarized degeneration family, hence does not quite fit into our notion of large complex structure limit. In our perspective, Gross-Wilson is an inspiration, rather than an example of the SYZ conjecture. which includes an almost explicit description of the metric, and a special Lagrangian torus fibration exists globally on $X$. It is partially generalized to higher dimensional hyperkähler manifolds with holomorphic Lagrangian abelian variety fibrations. After hyperkähler rotation, these can be regarded as special Lagrangian fibrations. Tosatti et al. [72] [31] established the semiflat metric asymptote in the generic region, and Gromov-Hausdorff collapse to the base manifold.

For general information on this section, see [42, Chapter 8,9], and the introduction in [54]. In the initial years following the SYZ proposal, there was an overly optimistic belief based on the analogy with the hyperkähler case, and based on topological and complex geometric considerations:

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  The special Lagrangian fibration exists globally and is defined by a $C^{\infty}$ map, even though some fibres may be singular.

• •

  The discriminant locus on the base is codimension two. For Calabi-Yau 3-folds, under suitable genericity assumption, the discriminant locus is a trivalent graph, with two types of vertices, known as positive and negative vertices.³³ 3 The names ‘positive/negative vertex’ come from some old fashioned topological models of the torus fibration where the most singular fibres have Euler characteristics $\pm 1$ respectively.

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  Along the edges of the trivalent graph, the singularity of the SYZ fibration is transversely modelled on the $I_{1}$ singularity.

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  The local region near the positive vertex is complex geometrically a large open subset inside

  $$\{z_{0}z_{1}z_{2}=1-z_{3}\}\subset\mathbb{C}^{3}\times\mathbb{C}^{*}_{z_{3}},\quad\Omega\propto\frac{1}{z_{3}}dz_{0}\wedge dz_{1}\wedge dz_{2},$$

  while the negative vertex region is modelled on a large open subset inside

  $$\{z_{3}z_{4}=1-z_{1}-z_{2}\}\subset(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}\times\mathbb{C}^{2}_{z_{3},z_{4}},\quad\Omega\propto\frac{1}{z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}.$$

The naïvete was challenged by Joyce [41], based on his observations concerning special Lagrangian singularities, which are markedly different from those visible in holomorphic fibrations. A global special Lagrangian fibration on the compact Calabi-Yau manifolds near the large complex structure, should it exist at all, is expected to have a much more subtle structure:

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  The special Lagrangian fibration is typically not defined by $C^{\infty}$ maps, but are at best piecewise smooth.

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  The discriminant locus on the base is typically not codimension two, but the trivalent graph is expected to be thickened to a codimension one ‘ribbon’. The amount of thickening probably tends to zero in the large complex structure limit.

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  The $I_{1}\times\mathbb{R}$ local singularity model does not lead to a Fredholm deformation theory for the singular special Lagrangian fibres, so should be replaced by some other singularity models.

Stepping aside from the substantial difficulties of the special Lagrangian local singularities, another major difficulty is to understand the Calabi-Yau metrics near the large complex structure limit, in complex dimension three. The optimistic expectations are inspired by the Gross-Wilson picture in complex dimension two. Hypothetically, the Calabi-Yau 3-fold is Gromov-Hausdorff close to a real 3-dimensional manifold, whose topology is believed to be the 3-sphere⁴⁴ 4 This expectation comes from the topology of the essential skeleton, see section 3.3 below., containing a trivialent graph, such that

• •

  Away from the trivalent graph, the Calabi-Yau metric is semiflat up to exponentially small errors.

• •

  Transverse to the edges in the trivalent graph, the metric is modelled on the Ooguri-Vafa metric appearing in Gross and Wilson’s picture.

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  Near the positive and negative vertices of the trivalent graph, the local metric is modelled on some generalization of the Ooguri-Vafa metrics.

It was recently realized that there exist almost canonical constructions of Ooguri-Vafa type metrics in complex dimension three, with the predicted topology and complex structure of the positive and negative vertices, constructed from a (nonlinear) generalized Gibbons-Hawking ansatz [54]. The asymptotic geometry of these Ooguri-Vafa type metrics matches with semiflat metrics in the generic region. The positive vertex metric contains a local region modelled on a generalized Taub-NUT type metric on $\mathbb{C}^{3}$, analogous to the way the Ooguri-Vafa metric contains a region modelled on the Taub-NUT metric.

The following questions are widely open:

Due to the analytic difficulty of the SYZ conjecture, especially the nongeneric regions with large Riemannian curvature, the literature has developed many interpretations of the conjectures, with somewhat diverging goals.

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  (Soft versions) For applications to homological mirror symmetry, one is primarily interested in constructing Lagrangian fibrations without requiring $\text{Im}(\Omega)|_{L}=0$, and therefrom build a mirror manifold $X^{\vee}$ and prove the categorical predictions $D^{b}Coh(X^{\vee})\simeq D^{\pi}Fuk(X,\omega)$. This viewpoint separates the symplectic and holomorphic data of the Calabi-Yau manifold, and has a topological/algebraic flavour.

  002H

  Remark 1. The special Lagrangian condition usually left out of homological mirror symmetry discussions, is supposedly related to Bridgeland stability conditions, which is a popular categorical interpretation of the BPS condition on D-branes.

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  (Strong metric version) On compact Calabi-Yau manifolds near the large complex structure limit, find a global special Lagrangian torus fibration.

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  (Weak metric version) Prove for a suitable class of compact Calabi-Yau manifolds near the large complex structure limit that a special Lagrangian $T^{n}$-fibration exists on a large subset with at least $99\%$ of the Calabi-Yau measure. More precisely, the percentage converges to $100\%$ in the limit.

The metric versions are highly sensitive to the Calabi-Yau metric, which is a much more rigid structure compared to the soft versions. They conform to the PDE spirit of the SYZ paper, while the soft versions are closer to the mirror symmetry motivations of the SYZ conjecture.

The strong metric version is perhaps the most faithful to the original intention of the SYZ paper. Its main evidence comes from the special case of hyperkähler metrics, mentioned in section 2.3. Other peripheral evidence comes from the construction of Lagrangian fibrations in many examples, ignoring the $\text{Im}(\Omega)|_{L}=0$ condition [58]. There are however many subtleties besetting this strong version, mentioned in section 2.4, making the conjecture very formidable, and by comparison the supporting evidence seems inadequate. Notably, the special Lagrangian singularities are not sufficiently understood, and we are not aware of any argument that definitively rules out special Lagrangians intersecting each other in the non-generic regions with large Riemannian curvature. As food for future thought, a somewhat weakened version bypassing these possible objections is

The weak metric version, on the other hand, concerns only the generic region, which is more accessible than the strong version. It conforms to the more cautious expectation, that the special Lagrangian fibration is only a limiting phenomenon. The bulk of the survey will focus on this weak version.

Consider an algebraic Calabi-Yau degeneration family $X\to S\setminus\{0\}$ as above. We follow [3] to consider the asymptote of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$ as $t\to 0$. Along the way, we will introduce the concept of dual intersection complexes and essential skeletons, which are simplicial complexes encoding the intersection patterns of divisors on the central fibre. An important lesson is that the measure theoretic limit of the Calabi-Yau manifolds is closer to simplicial complexes than algebraic varieties, indicating that the metric limit must be significantly different from Fubini-Study metrics associated to projective embeddings of bounded degree.

A very useful tool is to fill in the central fibre by choosing an snc model (cf. Remark 2) $\mathcal{X}$ over $S$. The central fibre $\mathcal{X}_{0}$ is an snc divisor with components $E_{i}$ for $i\in I$, and we write $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$. In the special case of semistable snc models $b_{i}=1$ for $i\in I$; this can always be achieved after finite base change. The canonical divisor $K_{\mathcal{X}}$ is supported on $\mathcal{X}_{0}$ as $K_{X}$ has a trivialising section $\Omega$. We may write $K_{\mathcal{X}}=\sum_{i}(a_{i}+b_{i}-1)E_{i}$, so that the relative log canonical divisor

Shifting all $a_{i}$ by a constant $\kappa$ is equivalent to multiplying $\Omega$ by $t^{\kappa}$, which gives an elementary factor $|t|^{2\kappa}$ to $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. Thus we shall always assume $\min a_{i}=0$.

It is useful to introduce a quantitative stratification on $X_{t}$ according to the intersection pattern of $E_{i}$. Let $E_{J}=\cap_{i\in J}E_{i}$ for $J\subset I$, which is irreducible if nonempty. Using the distance function of a fixed smooth background Kähler metric on $\mathcal{X}$, we can write

Around $\emptyset\neq E_{J}\subset\mathcal{X}$, we denote $p=|J|-1$, and introduce local coordinates $z_{0},\ldots z_{n}$ on $\mathcal{X}$, such that $z_{0},z_{1},\ldots,z_{p}$ are the defining equations of $E_{i}$ for $i\in J$. The conditions on the divisors mean that away from deeper strata we may arrange $t=\prod_{0}^{p}z_{i}^{b_{i}}$, and

for some local nowhere vanishing holomorphic function $u_{J}$. By definition $\Omega=dt\wedge\Omega_{t}$ along $X_{t}$, so on $E_{J}^{0}$

Notice also that the local equation $t=\prod_{0}^{p}z_{i}^{b_{i}}$ has $b_{J}=\gcd_{i\in J}b_{i}$ sheets of solutions. Using the polar coordinates by $z_{i}=e^{x_{i}\log|t|+\sqrt{-1}\theta_{i}}$ for $i\in J$, ones sees that the magnitude of $\int_{E_{J}^{0}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}$ is $O(|\log|t||^{l})$ for $l=|\{j\in J:a_{j}=0\}|-1$.

The local logarithmic variables $x_{i}=\frac{\log|z_{i}|}{\log|t|}$ lie on the simplex

These depend on the choice of $z_{i}$, but since the local defining equation of divisors differ by a nowhere vanishing holomorphic function, the ambiguity of $x_{i}$ is only $O(\frac{1}{|\log|t||})$ for $0<|t|\ll 1$. Taking a more global viewpoint, the combinatorial pattern of how these simplices fit together exactly reflects the intersection pattern of the divisors $E_{i}$. Formally, this information is encoded in the dual intersection complex $\Delta_{\mathcal{X}}$ for the snc model $\mathcal{X}$: this is the polyhedral complex whose vertices $v_{i}$ correspond to $E_{i}$, and we assign a simplex $\Delta_{J}$ with vertices $v_{i}$ for $i\in J$ if and only if $E_{J}\neq 0$. The coodinates $x_{j}$ then define a piecewise integral affine structure on $\Delta_{\mathcal{X}}$. Up to the above $O(\frac{1}{|\log|t||})$ ambiguity, we now have a logarithm map $\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$, locally described by $x_{i}=\frac{\log|z_{i}|}{\log|t|}$. Consequently, the ‘hybrid’ space $X\sqcup\Delta_{\mathcal{X}}$ is equipped with a natural topology, so that a sequence of points $z_{k}\in X_{t}$ converges to $x\in\Delta_{\mathcal{X}}$ iff $t\to 0$ and $\text{Log}_{\mathcal{X}}(z_{k})\to x$. The name ‘hybrid’ refers to the mixture of algebraic varieties with simplicial objects, which is better suited for measure theoretic limits, than the algebraic family $\mathcal{X}$.

The measure also singles out a distinguished subcomplex $Sk(\mathcal{X})$, called the essential skeleton, consisting of the simplices in $\Delta_{\mathcal{X}}$ whose vertices correspond to $E_{i}$ with $a_{i}=0$. This is where the limit of the normalised CY measure is supported. The dimension of $Sk(\mathcal{X})$ is a measurement of how transcendental the degeneration $X$ is; it is reflected by the growth order of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. The largest possible value for the dimension is $n$.

In the case of a large complex structure limit, $\dim_{\mathbb{R}}Sk(\mathcal{X})=n$. Let us analyze the CY measure more explicitly, in a semistable snc model. For $E_{J}$ corresponding to an $n$-dimensional simplex in $Sk(\mathcal{X})$, on $E_{J}^{0}$

Here $u_{J}$ limits to its value $u_{J}(E_{J})$ at the point stratum $E_{J}$, which is called the Poincaré residue of $\Omega$, and is easily seen to be independent of the choice of coordinates $z_{i}$. It is a consequence of the residue theorem on Riemann surfaces that $|u_{J}(E_{J})|^{2}$ is independent of such $J$ [3, Thm. 7.1]. Thus the pushforward to $\Delta_{\mathcal{X}}$ of the normalised CY measure (5) converges smoothly in the interior of $\Delta_{J}$ to a constant multiple of the Lebesgue measure:

Notice $dx_{1}\ldots dx_{n}$ is canonically defined due to the presence of an integral affine structure on $\Delta_{J}$. Viewed as a measure on $\Delta_{\mathcal{X}}$, the limit $d\mu_{0}$ has null measure on the complement of the $n$-dimensional faces of $Sk(\mathcal{X})$, as the integral of $d\mu_{t}$ in the corresponding region is $O(\frac{1}{|\log|t||})$. The constant in (8) is independent of $J$ and its sole purpose is to make $d\mu_{0}$ a probability measure.

A major caveat is that snc models are highly non-unique, because one can always blow up a given snc model along some locus inside the central fibre. The intrinsic information concerning polarized degenerations, such as the limiting behaviour of metrics and volume measures, are independent of particular snc models.

The effect of blow up on the dual intersection complex and essential skeleton is well understood (cf. [49, A.4]). The intuitive picture is as follows:

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  If we blow up an snc model $\mathcal{X}$ along a smooth irreducible subvariety properly contained inside $E_{J}=\cap_{j\in J}E_{j}$, but not contained in any smaller intersection stratum, then the new dual intersection complex contains $\Delta_{\mathcal{X}}$, while introducing a new vertex, and some new wings over certain faces of $\Delta_{\mathcal{X}}$. The essential skeleton remains intact.

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  If we blow up an snc model $\mathcal{X}$ along some $E_{J}=\cap_{j\in J}E_{j}$, then the new dual intersection complex is a subdivision of $\Delta_{\mathcal{X}}$. If the simplex $\Delta_{J}$ corresponding to $E_{J}$ is a face of the essential skeleton $Sk(\mathcal{X})$, then there is an induced subdivision on $Sk(\mathcal{X})$, and otherwise the essential skeleton remains intact.

While $\Delta_{\mathcal{X}}$ generally becomes larger under blow ups, the essential skeleton $Sk(\mathcal{X})$ is only subdivided, and its piecewise affine structure (in particular its homeomorphism type) is a birational invariant, denoted as $Sk(X)$. The birational invariance is not surprising: the essential skeleton is the measure theoretic limit of $X_{t}$, a property independent of the choice of models.

In an attempt to extract limiting information from the SYZ picture, Kontsevich and Soibelman [48][49] proposed the following picture. Let $X_{t}$ be a polarized degeneration of Calabi-Yau manifolds near the large complex structure limit, then

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  (Kähler class normalization) The rescaled metrics $\omega_{CY,t}\in\frac{1}{|\log|t||}c_{1}(\mathcal{O}(1))$ have nontrivial finite diameter Gromov Hausdorff subsequential limits.

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  There is an affine structure on the essential skeleton $Sk(X)$ away from a Hausdorff codimension two singular subset. On the smooth locus, there is a Riemannian metric obtained as the Hessian of local solutions to the real Monge-Ampère equation. This metric agrees with the Gromov-Hausdorff limit, which is conjecturally independent of the subsequence.

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  (Topology) Under the strict Calabi-Yau condition $h^{p,0}(X_{t})=0$ for $p<n$, the essential skeleton is homeomorphic to $S^{n}$.

The heuristic idea is that the essential skeleton should be the base of the hypothetical SYZ fibration, and the SYZ fibration is approximated by logarithm maps, at least in the generic region. We briefly comment on the status of the Kontsevich-Soibelman conjecture (cf. also [53, section 4.5]):

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  The uniform diameter estimate independent of small $t$

  $$C^{-1}\leq\text{diam}(X,\omega_{CY,t})\leq C,$$

  is recently established in joint work with Tosatti [56], using primarily Riemannian geometric methods. This together with Gromov compactness proves the Kähler class normalization prediction.

• •

  The prediction about the existence of a metrically preferred affine structure away from codimension two on the base, is part of the general lore of the SYZ conjecture (cf. section 2.4), even though there is insufficient evidence. Generally speaking, the simplicial complex structure on $Sk(X)$ induces a piecewise affine structure, and improving it to an affine structure away from codimension two, would require highly nontrivial choices. The author is not aware of a completely satisfactory answer to the following elementary question:

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  Question 5. Given a one-parameter family of quartic K3 surfaces near the large complex structure limit, without special symmetry, how can we determine the location of singular points on $Sk(X)$? How can we write down the affine structure on the regular locus explicitly?

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  The topology of the essential skeleton is an active research topic in birational geometry. The homeomorphism between $Sk(X)$ with $S^{n}$ is verified for many examples. In general, it is known [62][63] that $Sk(X)$ is a ‘pseudomanifold’, its rational homology groups agree with $S^{n}$, and its fundamental group has trivial profinite completion, but the actual homeomorphism type is still elusive, and supposedly requires appealing to the Poincaré conjecture.

A cornerstone of Kähler geometry is Yau’s celebrated proof of the Calabi conjecture, which implies

We now give a very sketchy account of (the modern view on) Yau’s proof of Theorem 4.1. Fix a background Kähler metric $\omega$ in the given class, and the task is to find a Kähler potential $\phi$ solving the complex Monge-Ampère equation for any given density function $e^{f}$ satisfying the cohomological constraint $\int_{X}e^{f}\omega^{n}=\int_{X}\omega^{n}$,

In particular for $e^{f}\omega^{n}=\text{const}\cdot\Omega\wedge\overline{\Omega}$ one obtains the Calabi-Yau metric. The uniqueness follows from a simple integration by parts argument.

The existence proof follows the continuity method, namely to deform through the space of Kähler metrics as $f$ deforms from zero to the desired function. Viewing (9) as defining a map from the potential to the density function, we need to show the map is submersive and proper. The submersion property is a consequence of standard Hodge theory on the Laplacian. To show properness one needs a priori estimates on $\phi$ depending only on $X$ and bounds on $f$, and for that matter we are free to impose any order of regularity on $f$.

The first step is to bound $\left\lVert\phi\right\rVert_{C^{0}}$ (known as ‘$C^{0}$-estimate’), using a technique called Moser iteration. We rewrite the equation as

For any $p>1$, multiply the equation by $\phi|\phi|^{p-2}$ and integrate by parts,

The key is that a weaker norm on RHS controls a stronger norm on LHS. This is reverse to the direction of the Poincaré inequality

and the Sobolev inequality

The combined force is that $\left\lVert\phi\right\rVert_{L^{2}_{1}}\leq C$, and one can inductively bound $\left\lVert\phi\right\rVert_{L^{p}}$ for large $p$, which turns out to be uniform in $p$. Taking the limit $p\to\infty$, this gives a bound $\left\lVert\phi\right\rVert_{L^{\infty}}\leq C(X,\left\lVert f\right\rVert_{L^{\infty}})$.

The second step is to bound $\left\lVert dd^{c}\phi\right\rVert_{C^{0}}$ (known as $C^{1,\bar{1}}$-estimate), i.e. to prove a uniform equivalence between $\omega$ and $\omega_{\phi}$. The starting point is a differential inequality by local calculations (in the analyst’s convention of Laplacian)

Subtracting a large enough multiply of the identity

we get a differential inequality

Using the a priori bound on $\phi$, an application of maximum principle then shows $\Tr_{\omega}\omega_{\phi}\leq C$.

The third step is to get higher order estimates. Standard elliptic theory implies it is sufficient to have a $C^{2,\alpha}$ bound on $\phi$. Evans-Krylov theory bridged the gap between $C^{1,\bar{1}}$-bound and $C^{2,\alpha}$-bound (cf. [66] Chapter 2, Section 4 for details). This argument is of local nature, and crucially uses that $\log\det(\partial_{i}\partial_{\bar{j}}\phi)$ is a concave function of the matrix $(\partial_{i}\partial_{\bar{j}}\phi)$, and the main tool is a Harnack inequality. The geometric insight is that higher order regularity is a manifestation of the local Euclidean nature of Kähler manifolds.

Yau’s proof strategy is highly influential and permeates the vast majority of works on CY metrics. The brief sketch above, however, highlights two reasons why it is difficult to adapt to the setting of SYZ conjecture:

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  The CY metrics undergoing large complex structure limit are highly degenerate, to the extent that the Sobolev constant becomes too big, so that the Moser iteration technique does not give useful $C^{0}$-estimate on the potential.

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  The $C^{1,\bar{1}}$ bound has global nature, so in order to obtain useful estimates on the Calabi-Yau metric only in the generic region, the design of the maximum principle must hold globally. This is hard on highly degenerate manifolds, where Riemannian curvature cannot be globally uniformly bounded.

For these reasons, our approach to the SYZ conjecture has significant departure from Yau’s proof. The potential estimates use complex pluripotential theory instead, and the metric estimate in the generic region uses a deep theorem of Savin in elliptic PDE theory, bypassing Yau’s estimates.

Complex pluripotential theory concerns the study of weak notions of Kähler potentials, and their wider implications on algebraic geometry and PDEs. They can be viewed as the generalization of potential theory on Riemann surfaces to several complex variables. Some common themes include:

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  The weak compactness theory for the space of potentials. This is suited for variational methods in Kähler geometry.

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  The relations between potential theory and algebraic geometry, via Hodge theory, $\bar{\partial}$-operator, intersection theory, etc. This provides transcendental methods to birational geometry.

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  The weak formulation of the complex Monge-Ampère equation, and a priori potential estimates under weak assumptions on the volume density, or in the presence of singularities for the ambient complex variety. This is useful for studying singularity formation of canonical metrics.

Unlike Yau’s proof of the Calabi conjecture, whose techniques are typical of elliptic PDEs, complex pluripotential theory is more akin to complex analysis, and frequently provides stronger results. To build up the intuition, we will first review the standard versions in the literature, before stating our uniform versions, which are foundational to our approach to the SYZ conjecture.

A basic problem in pluripotential theory is to estimate Kähler potentials. Kolodziej pioneered a method to achieve the following effects. The basic versions of his theorems work on a fixed ambient Kähler manifold $(Y,\omega)$, and deal with Kähler potentials $\phi\in PSH(Y,\omega)$ normalized to $\sup_{Y}\phi=0$.

• •

  (‘Potential estimate’) If the volume density of $\omega_{\phi}^{n}$ has some integrability control such as an $L^{p}$-bound

  $$\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\leq C,\quad p>1,$$

  (10)

  then $\phi$ has an $L^{\infty}$ bound depending only on $(X,\omega),n,p,C$ [45][20][24]. In fact this can be improved to a $C^{\alpha}$ Hölder bound on $\phi$ [47]. (Notice $p=1$ would not suffice, as the $L^{1}$-bound $\int_{Y}\omega_{\phi}^{n}\leq\int_{Y}\omega^{n}$ is automatic).

• •

  (‘$L^{1}$-stability estimates’) Suppose $\phi,\psi\in PSH(Y,\omega)$ are both subject to the $L^{p}$ volume density integrability control (10), and the normalization $\sup_{Y}\phi=\sup_{Y}\psi=0$. If

  1. 1.

     Either $\phi,\psi$ are close together in the $L^{1}$-sense $\int_{Y}|\psi-\phi|\omega^{n}\ll 1$, (‘$L^{1}$-potential stability’)
     

  2. 2.

     Or the volume densities of $\omega_{\phi}^{n}$ and $\omega_{\psi}^{n}$ are close together in the $L^{1}$-sense, namely the total variation $\int_{Y}|\omega_{\phi}^{n}-\omega_{\psi}^{n}|\ll 1$, (‘$L^{1}$-volume stability’)

  Then $|\phi-\psi|$ is small in the $L^{\infty}$ sense with quantitative estimates [46].

In our applications, we need to work with a family of Kähler manifolds, and the estimates need to be uniform under very severe complex structure degenerations, and allowing the total volume to collapse to zero. Some subtleties are:

• •

  The Calabi-Yau volume measure is very different from the volume form of a Fubini-Study metric (cf. section 3.1), so the idea of volume density with respect to a Fubini-Study style ambient metric is no longer appropriate. The replacement of (10) turns out to be a Skoda type estimate (11).

• •

  Unlike $L^{\infty}$ bounds, the Hölder norm depends strongly on the choice of the ambient metric, which specifies a choice of distance function. We think it is highly non-obvious how to make a semi-explicit choice uniformly in the family, and therefore we do not attempt to generalize Hölder estimates.

• •

  A technical problem in our applications involving the comparison of two potentials, only one potential has volume density control. Thus unlike the $L^{1}$-stability estimates above, our version treats the two potentials asymmetrically.

Our analogue of the potential estimate is

The strength of this result is that it still applies when the complex/Kähler structures are highly degenerate, as it distills the dependence on $(Y,\omega)$ to only 3 constants $n,\alpha,A$. The relations to the more standard version above can be explained as follows:

• •

  For a fixed ambient Kähler metric, by the Hölder inequality and the standard Skoda inequality Theorem 4.3, under the density integrability assumption (10), then for $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, and some small enough $\beta>0$,

  $$\int_{Y}e^{-\beta u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq\frac{1}{\text{Vol}(Y)}\left(\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\omega^{n}\right)^{1/p}\left(\int_{Y}e^{-p^{\prime}\beta u}\omega^{n}\right)^{1/p^{\prime}}\leq\text{const}.$$

  This means the Skoda type estimate (11) is a weaker assumption than (10), even in the standard setting.

• •

  The first part of the conclusion recovers Kolodziej’s potential estimate.

• •

  The second part of the conclusion is about comparing the two potentials $\phi$ versus $-t_{0}$. The condition $\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ means $\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}$ is small, which by the Hölder inequality and the density integrability (10) follows from the smallness of $\int_{\phi\leq-t_{0}}\omega^{n}$. This should be viewed as one half of the $L^{1}$-potential stability condition $\int_{Y}|\phi+t_{0}|\omega^{n}\ll 1.$ The conclusion for the lower bound on $\min\phi$, should be viewed as one half of a smallness bound on the $C^{0}$-norm of $\phi+t_{0}$, namely the two potentials $\phi$ and $-t_{0}$ are close together in $C^{0}$-norm.

• •

  To generalize to the cases with $\psi$ not necessarily constant, it suffices to replace $\omega$ by $\omega_{\psi}$, and $\phi$ by $\phi-\psi$ in Theorem 4.5.

Combining Thm. 4.4 with Thm. 4.5, we immediately obtain

Our adaption of the $L^{1}$-volume stability estimate is