List of symbols [02YQ]

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List of symbols

$A$

affine map of vector spaces, \hyperpage34
$A_{d}(X)$

Chow group of $d$ -dimensional cycles, \hyperpage59
$\mathscr{A}^{\ast}_{X^{{\text{\rm an}}}}$

sheaf of differential forms of a complex space, \hyperpage12
$\operatorname{aff}(C)$

affine hull of a convex set, \hyperpage25
$C_{x}$

piece of a convex decomposition, \hyperpage30
$C(\Delta,u,V)$

polynomial associated to an aggregate, \hyperpage112
$C_{k}(\Delta,u,V)$

coefficient of $C(\Delta,u,V)$ , \hyperpage112
$\operatorname{c}_{1}({\overline{L}})$

Chern form of a smooth metrized line bundle, \hyperpage12
$\operatorname{c}_{1}({\overline{L}}_{0})\wedge\cdots\wedge\operatorname{c}_{1}({\overline{L}}_{d-1})\wedge\delta_{Y}$

signed measure (smooth case), \hyperpage13
$\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\land\delta_{Y}$

signed measure (algebraic case), \hyperpage19
$\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\land\delta_{Y}$

signed measure (integrable case), \hyperpage20
$\operatorname{c}(C)$

cone of a convex set, \hyperpage25
$\operatorname{c}(\Pi)$

cone of a polyhedral complex, \hyperpage27
$C^{\infty}_{X^{{\text{\rm an}}}}$

sheaf of smooth functions of a complex space, \hyperpage12
${\operatorname{cl}}(f)$

closure of a concave function, \hyperpage29
$\operatorname{cone}(b_{1},\dots,b_{l})$

cone generated by a set of vectors, \hyperpage27
$\operatorname{conv}(b_{1},\dots,b_{l})$

convex hull of a set of points, \hyperpage27
$\mathbb{D}$

unit disk of $\mathbb{C}$ , \hyperpage12
${\mathscr{D}(\Lambda)}$

space of differences of piecewise affine concave functions, \hyperpage45
${\overline{\mathscr{D}(\Lambda)}}$

space of differences of uniform limits of piecewise affine concave functions, \hyperpage45
$D$

Cartier divisor, \hyperpage56
$D_{\Psi}$

$\mathbb{T}$ -Cartier divisor on a toric variety, \hyperpage57
$D_{\psi}$

$\mathbb{T}$ -Cartier divisor on a toric scheme, \hyperpage69
$\operatorname{def}(\alpha)$

defect of the product formula on an adelic field, \hyperpage23
$\operatorname{Div}_{\mathbb{T}}(X_{\Sigma})$

group of $\mathbb{T}$ -Cartier divisors, \hyperpage59
${\operatorname{dom}}(f)$

effective domain of a concave function, \hyperpage29
${\operatorname{dom}}(\partial f)$

effective domain of the sup-differential, \hyperpage30
$E_{i},F_{i,j}$

components of the special fibre of a semi-stable model of $\mathbb{P}^{1}$ , \hyperpage93
$e_{H/K}$

ramification degree of a finite field extension, \hyperpage91
${\operatorname{\mathbf{e}}}_{K}$

parameterization of a variety with corners, \hyperpage80
$F_{\sigma}$

face dual to a cone, \hyperpage42
$g\circ\partial f$

a ${\mathcal{M}}(f)$ -measurable function, \hyperpage50
$g_{{\overline{L}},s}$

function on $X_{0}^{{\text{\rm an}}}$ associated to a metrized toric line bundle and a toric section, \hyperpage81
$H$

linear map of lattices, \hyperpage54
$H$

linear map of vector spaces, \hyperpage34
$H_{i}$

standard hyperplane of $\mathbb{P}^{n}$ , \hyperpage59
$\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})$

local height, \hyperpage21
$\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)$

global height, \hyperpage25
$\mathscr{H}(p)$

field associated to a point of a Berkovich space, \hyperpage14
$\operatorname{hypo}(f)$

hypograph of a concave function, \hyperpage31
$\mathbb{K}$

adelic field, \hyperpage23
$\mathbb{K}_{v}$

completion of an adelic field, \hyperpage23
$\mathbb{K}^{\circ}_{S}$

subring of an adelic field, \hyperpage24
$K$

non-Archimedean field, \hyperpage13
$K^{\circ}$

valuation ring of a non-Archimedean field, \hyperpage13
$K^{\circ\circ}$

maximal ideal of a valuation ring, \hyperpage13
$k$

residue field of a valuation ring, \hyperpage13
$K[M_{\sigma}]$

semigroup $K$ -algebra of a cone, \hyperpage52
$K^{\circ}[{\widetilde{M}}_{\sigma}]$

semigroup $K^{\circ}$ -algebra of a cone, \hyperpage64
$K^{\circ}[{\mathcal{X}}_{\sigma}]$

ring of functions of an affine toric scheme, \hyperpage64
$K^{\circ}[{\widetilde{M}}_{\Lambda}]$

semigroup $K^{\circ}$ -algebra of a polyhedron, \hyperpage65
$K^{\circ}[{\mathcal{X}}_{\Lambda}]$

ring of functions of an affine toric scheme, \hyperpage65
$\mathcal{K}_{X}$

sheaf of rational functions of a scheme, \hyperpage52
${\mathcal{L}}f$

Legendre-Fenchel correspondence of a concave function, \hyperpage32
${\mathcal{L}}$

model of a line bundle, \hyperpage16
$L$

line bundle, \hyperpage12
${\overline{L}}$

metrized line bundle (Archimedean case), \hyperpage12
${\overline{L}}$

metrized line bundle (non-Archimedean case), \hyperpage15
$L_{\Psi}$

toric line bundle determined by a virtual support function, \hyperpage58
$L^{{\text{\rm an}}}$

analytification of a line bundle over $\mathbb{C}$ , \hyperpage12
$L^{{\text{\rm an}}}$

Berkovich analytification of a line bundle, \hyperpage15
${\overline{L}}^{{\operatorname{can}}}$

toric line bundle with its canonical metric, \hyperpage83
$L_{\Lambda}$

linear space associated to a polyhedron, \hyperpage49
$m_{\sigma}$

defining vector of a virtual support function, \hyperpage57
${\mathcal{M}}_{\mu}(f)$

Monge-Ampère measure associated to a concave function and a measure, \hyperpage47
${\mathcal{M}}_{M}(f)$

Monge-Ampère measure associated to a concave function and a lattice, \hyperpage49
${\mathcal{M}}_{M}(f_{1},\dots,f_{n})$

mixed Monge-Ampère measure, \hyperpage50
${\overline{\mathcal{M}}}_{M}(\psi)$

Monge-Ampère measure on $N_{\Sigma}$ , \hyperpage87
$\operatorname{MV}_{M}(Q_{1},\dots,Q_{n})$

mixed volume of a family of convex sets, \hyperpage51
$\operatorname{MI}_{M}(g_{0},\dots,g_{n})$

mixed integral of a family of concave functions, \hyperpage51
$\mathfrak{M}_{\mathbb{K}}$

absolute values and weights of an adelic field, \hyperpage23
$M$

lattice dual to $N$ , \hyperpage28
$M_{\mathbb{R}}$

vector space dual to $N_{\mathbb{R}}$ , \hyperpage25
${\widetilde{M}}$

$M\oplus\mathbb{Z}$ , \hyperpage64
$M_{\sigma}$

semigroup of $M$ associated to a cone, \hyperpage52
$M(\sigma)$

dual sublattice associated to a cone, \hyperpage53
${\widetilde{M}}_{\sigma}$

semigroup of ${\widetilde{M}}$ associated to a cone, \hyperpage64
${\widetilde{M}}_{\Lambda}$

semigroup associated to a polyhedron, \hyperpage65
$M(\Lambda)$

sublattice associated to a polyhedron in $M_{\mathbb{R}}$ , \hyperpage49
$M(\Lambda)$

sublattice associated to a polyhedron in $N_{\mathbb{R}}$ , \hyperpage67
${\widetilde{M}}(\Lambda)$

sublattice of ${\widetilde{M}}$ associated to a polyhedron, \hyperpage66
$\operatorname{mult}(\Lambda)$

multiplicity of a polyhedron, \hyperpage67
$N$

lattice, \hyperpage28
$N_{\mathbb{R}}$

real vector space, \hyperpage25
${\widetilde{N}}$

$N\oplus\mathbb{Z}$ , \hyperpage64
$N(\sigma)$

quotient lattice associated to a cone, \hyperpage53
$N(\Lambda)$

quotient lattice of $N$ associated to a polyhedron, \hyperpage67
${\widetilde{N}}(\Lambda)$

quotient lattice of ${\widetilde{N}}$ associated to a polyhedron, \hyperpage66
$N_{\Sigma}$

compactification of $N_{\mathbb{R}}$ with respect to a fan, \hyperpage80
$n_{v}$

weight of an absolute value, \hyperpage23
$o$

special point of $S$ , \hyperpage15
$O(\sigma)$

orbit in a toric variety, \hyperpage53
$O(\Lambda)$

vertical orbit in a toric scheme, \hyperpage66
$\mathcal{O}_{X}$

sheaf of algebraic functions of a scheme, \hyperpage52
$\mathcal{O}_{X^{{\text{\rm an}}}}$

sheaf of analytic functions of a complex space, \hyperpage12
$\mathcal{O}_{X^{{\text{\rm an}}}}$

sheaf of analytic functions of a Berkovich space, \hyperpage14
$\mathcal{O}(D)$

line bundle associated to a Cartier divisor, \hyperpage58
$P$

lattice dual to $Q$ , \hyperpage56
$P_{f}$

pairing associated to a concave function, \hyperpage30
$\operatorname{Pic}(X)$

Picard group, \hyperpage59
$\mathscr{P}$

space of piecewise affine functions, \hyperpage43
$\mathscr{P}(\Lambda)$

space of piecewise affine functions with given effective domain, \hyperpage43
$\mathscr{P}(\Lambda,\Lambda^{\prime})$

space of piecewise affine functions with given effective domain and stability set, \hyperpage43
${\overline{\mathscr{P}}}$

closure of $\mathscr{P}$ , \hyperpage43
${\overline{\mathscr{P}(\Lambda)}}$

closure of $\mathscr{P}(\Lambda)$ , \hyperpage43
${\overline{\mathscr{P}}}(\Lambda,\Lambda^{\prime})$

closure of $\mathscr{P}(\Lambda,\Lambda^{\prime})$ , \hyperpage43
$\mathfrak{p}_{p}$

prime ideal of a point of a Berkovich space, \hyperpage14
$Q$

saturated sublattice of $N$ , \hyperpage55
${\underline{\mathbb{R}}}$

real line with $-\infty$ added, \hyperpage29
$\operatorname{rec}(\Pi)$

recession of a polyhedral complex, \hyperpage27
$\operatorname{rec}(C)$

recession cone of a convex set, \hyperpage25
$\operatorname{rec}(f)$

recession function of a concave function, \hyperpage36
$\operatorname{rec}(f)$

recession of a difference of concave functions, \hyperpage46
${\operatorname{red}}$

reduction map, \hyperpage16
$\operatorname{ri}(C)$

relative interior of a convex set, \hyperpage25
$S$

scheme associated to a DRV, \hyperpage15
$s_{\Psi}$

toric section determined by a virtual support function, \hyperpage58
$\mathbb{S}^{{\text{\rm an}}}$

compact torus, \hyperpage79
$\operatorname{stab}(f)$

stability set of a concave function, \hyperpage29
$\mathbb{T}$

split algebraic torus, \hyperpage52
$\mathbb{T}_{M}$

algebraic torus associated to a lattice, \hyperpage15
${\operatorname{val}}$

valuation map of a non-Archimedean field, \hyperpage68
${\operatorname{val}}_{K}$

valuation map on an analytic toric variety, \hyperpage80
$v_{\tau}$

smallest nonzero lattice point in a ray, \hyperpage58
$v_{F}$

integral inner orthogonal vector of a facet, \hyperpage49
$V(\sigma)$

closure of an orbit of a toric variety, \hyperpage53
${\mathcal{V}}(\sigma)$

horizontal closure of an orbit of a toric scheme, \hyperpage66
$V(\Lambda)$

vertical closure of a orbit of a toric scheme, \hyperpage67
$\operatorname{vol}_{L}$

normalized Haar measure, \hyperpage49
$X^{{\text{\rm an}}}$

analytification of a variety over $\mathbb{C}$ , \hyperpage11
$X^{{\text{\rm an}}}$

Berkovich space of a scheme, \hyperpage13
$X_{{\text{\rm alg}}}$

algebraic points of a variety, \hyperpage14
$X^{{\text{\rm an}}}_{{\text{\rm alg}}}$

algebraic points of a Berkovich space, \hyperpage14
$X^{\text{\rm an}}(K)$

rational points of a Berkovich space, \hyperpage14
$X_{\sigma}$

affine toric variety, \hyperpage52
$X_{\Sigma}$

toric variety associated to a fan, \hyperpage52
$X_{\Sigma,0}$

principal open subset, \hyperpage52
$X_{\Sigma}(\mathbb{R}_{\geq 0})$

variety with corners associated to a toric variety, \hyperpage78
${\mathcal{X}}$

model of a variety, \hyperpage15
$\mathcal{X}_{o}$

special fiber of a scheme over $S$ , \hyperpage15
$\mathcal{X}_{\eta}$

generic fiber of a scheme over $S$ , \hyperpage15
${\mathcal{X}}_{\sigma}$

affine toric scheme associated to a cone, \hyperpage64
${\mathcal{X}}_{\Lambda}$

affine toric scheme associated to a polyhedron, \hyperpage65
${\mathcal{X}}_{{\widetilde{\Sigma}}}$

toric scheme associated to a fan, \hyperpage65
${\mathcal{X}}_{\Pi}$

toric scheme associated to a polyhedral complex, \hyperpage66
$(\mathcal{X},\mathcal{L},e)$

model of a variety and a line bundle, \hyperpage16
${\mathcal{X}}_{\mathbb{S}}$

toric model associated to a semi-stable model, \hyperpage93
$x_{\sigma}$

distinguished point of an affine toric variety, \hyperpage53
$Y_{\Sigma,Q}$

toric subvariety, \hyperpage56
$Y_{\Sigma,Q,p}$

translated toric subvariety, \hyperpage56
$z_{\Psi}$

toric structure determined by a virtual support function, \hyperpage58
$Z_{n-1}^{\mathbb{T}}(X_{\Sigma})$

group of $\mathbb{T}$ -Weil divisors, \hyperpage59
$\Delta_{\Psi}$

polytope associated to a virtual support function, \hyperpage61
$\Delta^{n}$

standard simplex, \hyperpage37
$\eta$

generic point of $S$ , \hyperpage15
$\Theta_{i}$

set of components of the special fibre of a semi-stable model of $\mathbb{P}^{1}$ , \hyperpage93
$\vartheta_{{\overline{L}},s}$

roof function, \hyperpage104
$\theta_{\Sigma}$

injection of the variety with corners in the corresponding analytic toric variety, \hyperpage79
$\iota$

inclusion of a saturated sublattice, \hyperpage55
$\iota_{C}$

indicator function of a convex set, \hyperpage29
$\iota_{\sigma}$

closed immersion of the closure of an orbit into a toric variety, \hyperpage54
$\lambda_{K}$

scalar associated to a local field, \hyperpage80
$\mu$

Haar measure on $M_{\mathbb{R}}$ , \hyperpage47
$\mu$

action of a torus on a toric variety, \hyperpage52
$\mu$

moment map, \hyperpage80
$\xi_{V}$

point associated to an irreducible component of the special fiber, \hyperpage16
$\varpi$

generator of the maximal ideal of a DVR, \hyperpage15
$\pi$

map from $X^{{\text{\rm an}}}$ to $X$ , \hyperpage14
$\pi_{\sigma}$

projection associated to a cone, \hyperpage53
$\Pi$

polyhedral complex, \hyperpage27
$\Pi^{i}$

set of $i$ -dimensional polyhedra of a complex, \hyperpage27
$\Pi(f)$

convex decomposition associated to a concave function, \hyperpage30
$\Pi(\sigma)$

star of a cone in a polyhedral complex, \hyperpage66
$\rho$

morphism of tori, \hyperpage54
$\rho_{H}$

morphism of tori induced by $H$ , \hyperpage54
$\rho_{\Sigma}$

projection of a toric variety to its associated variety with corners, \hyperpage78
$\sigma$

anti-linear involution defined by a variety over $\mathbb{R}$ , \hyperpage13
$\sigma$

cone, \hyperpage26
$\sigma_{F}$

cone dual to a face, \hyperpage42
$\Sigma$

fan, \hyperpage27
$\Sigma^{i}$

set of $i$ -dimensional cones of a fan, \hyperpage27
$\Sigma$

rational fan, \hyperpage52
$\Sigma_{\Delta}$

fan associated to a polytope, \hyperpage41
$\Sigma(\sigma)$

star of a cone in a fan, \hyperpage54
${\widetilde{\Sigma}}$

fan in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$ , \hyperpage64
$\tau_{u_{0}}f$

translate of a concave function, \hyperpage34
$\Phi_{p,A}$

equivariant morphism of toric schemes, \hyperpage69
$\varphi_{H}$

toric morphism of toric varieties, \hyperpage55
$\varphi_{p,H}$

equivariant morphism of toric varieties, \hyperpage55
$\chi^{m}$

character of $\mathbb{T}$ , \hyperpage52
$\Psi_{C}$

support function of a convex set, \hyperpage29
$\Psi$

virtual support function, \hyperpage57
$\Psi(\sigma)$

virtual support function induced on a quotient, \hyperpage59
$\psi$

H-lattice function, \hyperpage69
$\psi_{{\overline{L}},s}$ , $\psi_{\|\cdot\|}$

function on $N_{\mathbb{R}}$ associated to a metrized toric line bundle and a section, \hyperpage82
$\psi_{\|\cdot\|}$

function on $N_{\mathbb{R}}$ associated to a non-necessarily toric metric, \hyperpage91
$\psi_{g}$

function on $N_{\mathbb{R}}$ associated to a rational function, \hyperpage90
$A^{\ast}f$

inverse image of a concave function by an affine map, \hyperpage34
$A_{\ast}g$

direct image of a concave function by an affine map, \hyperpage34
$C^{\ast}$

corresponding convex set in a dual decomposition, \hyperpage32
[D]

Weil divisor associated to a Cartier divisor, \hyperpage58
$E\cdot F$

intersection product of two $1$ -cycles on a surface, \hyperpage93
$(\iota\cdot\operatorname{div}(s))$

intersection number of a curve with a divisor, \hyperpage17
$\Pi_{1}\cdot\Pi_{2}$

complex of intersections, \hyperpage28
$\angle(K,\Lambda)$

angle of a polyhedron at a face, \hyperpage41
$\sigma^{\bot}$

orthogonal space, \hyperpage53
$\sigma^{\vee}$

dual of a convex cone, \hyperpage41
$f^{\vee}$

Legendre-Fenchel dual of a concave function, \hyperpage29
$H^{\vee}$

dual of a linear map, \hyperpage34
$\lambda f$

left scalar multiplication, \hyperpage34
$f\lambda$

right scalar multiplication, \hyperpage34
$\nabla f$

gradient of a differentiable function, \hyperpage30
$\partial f$

sup-differential of a concave function, \hyperpage30
$f_{1}\boxplus f_{2}$

sup-convolution of concave functions, \hyperpage33
$|\Pi|$

support of a convex decomposition, \hyperpage26
$|\cdot|_{v}$

absolute value of an adelic field, \hyperpage23
$\|\cdot\|$

metric on a line bundle (Archimedean case), \hyperpage12
$\|\cdot\|$

metric on a line bundle (non-Archimedean case), \hyperpage15
$\|\cdot\|_{{\operatorname{can}}}$

non-Archimedean canonical metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ , \hyperpage18
$\|\cdot\|_{{\operatorname{can}}}$

Archimedean canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ , \hyperpage20
$\|\cdot\|_{{\operatorname{can}}}$

canonical metric of a toric line bundle, \hyperpage83
$\|\cdot\|_{\operatorname{FS}}$

Fubini-Study metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ , \hyperpage12
$\|\cdot\|_{{\mathcal{X}},{\mathcal{L}},e}$

metric induced by a model, \hyperpage16
$\|\cdot\|_{\mathbb{S}}$

toric metric from a metric (Archimedean case), \hyperpage88
$\|\cdot\|_{\mathbb{S}}$

toric metric from a metric (non-Archimedean case), \hyperpage91

Index

adelic field Definition 2.47
- group of global heights of Definition 2.48
adelic toric metric Definition 5.84
affine hull of a convex set §3.1
aggregate of a polytope §7.1
algebraic metric Definition 2.17
- semipositive Definition 2.26
- signed measure associated to Definition 2.28
algebraic point of a Berkovich space Definition 2.6
algebraic variety over $\mathbb{R}$ Remark 2.5
- metrized line bundle on Remark 2.5
analytic function on a Berkovich space §2.2
analytification
- of a line bundle §2.1, §2.3
- of a variety over a non-Archimedean field §2.2
- of a variety over $\mathbb{C}$ §2.1
angle of a polyhedron at a face Definition 3.66
approachable metric Definition 2.31, Corollary 5.28
- over an adelic field item 1
approachable metrized line bundle Definition 2.31
Archimedean case §2.4
Berkovich space §2.2—§2.2
- algebraic point of Definition 2.6
- analytic function on §2.2
- field associated to a point of §2.2
- points of §2.2
- rational point of Definition 2.6
- Shilov boundary of §2.3
- topology of §2.2
Brion’s “short formula” §7.1
canonical metric
- of a toric line bundle Proposition-Definition 5.20, Example 5.42
- of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ Example 2.25, Example 2.32, item 1
canonical model
- of a $\mathbb{T}$ -Cartier divisor Definition 4.76, Example 5.42
- of a toric variety Definition 4.63
Chow group §4.3
closure of a concave function §3.2
compact torus Example 2.8, §5.1, §5.4
concave function §3.2
- closed §3.2
- of Legendre type Definition 3.51
- sup-differentiable §3.2
cone §3.1
- of a convex set §3.1
- of a polyhedral complex Definition 3.7
conical function Example 3.16
convex decomposition Definition 3.2
- complete Definition 3.2
- in a subset Definition 3.2
convex polyhedral cone Definition 3.3
convex polyhedron Definition 3.3
- angle at a face Definition 3.66
- lattice Definition 3.12
- rational Definition 3.12
convex set §3.1
- dimension of §3.1
convex subdivision Definition 3.2
- complete Definition 3.2
- in a subset Definition 3.2
DC function, see difference of concave functions
defect of an adelic field Definition 2.48
defining vectors
- of a piecewise affine function Definition 3.90
- of a virtual support function Definition 4.15
difference of concave functions §3.6
differentiable function §3.4
differential forms on a complex space §2.1
direct image of a concave function by an affine map Definition 3.42
distance between metrics §2.4
distinguished point of an affine toric variety §4.1
dual
- convex decompositions Definition 3.32
  - induced by a concave function Definition 3.34
- of a convex cone Definition 3.67
- polyhedral complexes Definition 3.68
DVR (discrete valuation ring) §2.3
effective domain
- of a concave function §3.2
- of sup-differential §3.2
entropy §7.2
equivariant morphism
- of toric schemes Definition 4.70
- of toric varieties Definition 4.7, §4.2, §4.2, §4.2
- Stein factorization of §4.2
essential minimum §8.1
face of a convex set Definition 3.1
- exposed Definition 3.1
facet Definition 3.1
fan Definition 3.6, §4
- complete §4.1
- rational Definition 3.13, §4
first Chern form §2.1
Fubini-Study metric Example 2.2, item 2
generic fiber of a scheme over $S$ §2.3
global field Definition 2.51
H-representation
- of a piecewise affine concave function §3.5
- of a polyhedron §3.1
height of cycles
- global Definition 2.56
- local Definition 2.39
- toric Definition 6.32
- toric local Definition 6.1
Hessian matrix §3.7
horizontal orbit in a toric scheme §4.5
horizontal scheme §4.5
hypograph of a concave function §3.2
- non-vertical face of §3.2
indicator function of a convex set Example 3.16, Example 3.65
integrable cycle Definition 2.53
integrable metric Definition 2.31
- over an adelic field item 1
- signed measure associated to Definition 2.34
inverse image of a concave function by an affine map Definition 3.42
left scalar multiplication §3.3
Legendre transform §3.2, §3.4
Legendre-Fenchel correspondence of a concave function Definition 3.31
Legendre-Fenchel dual of a concave function §3.2
Lipchitzian function §3.6
measure
- associated to a smooth metric §2.1
- associated to an algebraic metric Definition 2.28
- associated to an integrable metric Definition 2.34
metric
- adelic toric, see adelic toric metric
- algebraic, see algebraic metric
- approachable, see approachable metric
- canonical, see canonical metric
- Fubini-Study, see Fubini-Study metric
- induced by a model Definition 2.17
- on a line bundle over a non-Archimedean field Definition 2.10
- on a line bundle over an adelic field item 1
- on a line bundle over $\mathbb{C}$ Definition 2.1
- quasi-algebraic, see quasi-algebraic metric
- smooth, see smooth metric
- toric, see toric metric
- toric algebraic, see toric algebraic metric
metrized line bundle §2.4
- algebraic Definition 2.17
- approachable Definition 2.31
- integrable Definition 2.31
- on an algebraic variety over $\mathbb{R}$ Remark 2.5
mixed integral of a family of concave functions Definition 3.113
mixed volume of a family of compact convex sets Definition 3.109
model
- canonical, see canonical model
- of a line bundle Definition 2.16
  - proper Definition 2.16
- of a variety Definition 2.11
  - proper Definition 2.11
- semi-stable Definition 5.54
- toric, see toric model
moment map Remark 5.8
Monge-Ampère measure Definition 3.92, Definition 5.32
- mixed Definition 3.107
Monge-Ampère operator Definition 3.92
multiplicity of a polyhedron Definition 4.68
Nakai-Moishezon criterion
- for a toric scheme §4.7
- for a toric variety §4.4
non-Archimedean case §2.4
non-Archimedean field §2.2
normalized Haar measure Definition 3.102
orbit
- in a toric scheme §4.5
- in a toric variety §4.2
Picard group §4.3
piecewise affine concave function §3.5
- difference of uniform limits of §3.6
- H-lattice Definition 3.73
- rational Definition 3.73
- V-lattice Definition 3.73
piecewise affine function Definition 3.58
- H-lattice Definition 3.88
- on a polyhedral complex Definition 3.60
- rational Definition 3.88
- V-lattice Definition 3.88
polyhedral complex Definition 3.6
- compatible with a piecewise affine function Definition 3.60
- conic Definition 3.6
- lattice Definition 3.13
- of intersections Definition 3.10
- rational Definition 3.13
- regular Definition 3.60
- SCR (strongly convex rational) Definition 3.13
- strongly convex Definition 3.6
polytope Definition 3.3
- associated to a virtual support function §4.4
- lattice §4
principal open subset of a toric variety §4.1
product formula Definition 2.48
projective space Example 5.26
- as a toric scheme Example 4.62
- as a toric variety Example 4.3
proper intersection Definition 2.38
quasi-algebraic metric item 2
ramification degree of a finite field extension §5.4
rational point of a Berkovich space Definition 2.6
recession cone of a convex set §3.1
recession function
- of a concave function Definition 3.48
- of a difference of concave functions Definition 3.85
recession of a polyhedral complex Definition 3.7
reduction map §2.3
relative interior of a convex set §3.1
right scalar multiplication §3.3
roof function Definition 6.8
- rational Definition 6.8
semigroup algebra §4.1
Shilov boundary of a Berkovich space §2.3
smooth metric Definition 2.1
- positive Definition 2.3
- semipositive Definition 2.3, Proposition 5.29
- signed measure associated to §2.1
special fiber of a scheme over $S$ §2.3
stability set of a concave function §3.2
standard simplex Example 3.53, Example 3.65, Example 5.18
star
- of a cone in a fan §4.2
- of a cone in a polyhedral complex §4.5
- of a convex polyhedron in a polyhedral complex §4.5
strictly concave function §3.4
- on a polyhedral complex Definition 3.60
strongly convex polyhedron Definition 3.3
successive minima §8.1
sup-convolution of concave functions §3.3
sup-differential of a concave function §3.2
- image of §3.2
support function
- of a convex set Example 3.16, Example 3.65
- of the standard simplex Example 3.65
- on a fan Definition 4.15
- virtual, see virtual support function
support of a convex decomposition Definition 3.2
symplectic potential Remark 5.74
$\mathbb{T}$ -Cartier divisor
- on a toric scheme §4.6
  - semipositive Definition 4.96
- on a toric variety Definition 4.14
$\mathbb{T}$ -Weil divisor Definition 4.24, §4.6
toric curve §8.1
- height of §8.1
toric line bundle
- on a toric scheme Definition 4.77
  - toric section of Definition 4.77
- on a toric variety Definition 4.19
  - toric section of Definition 4.19
toric metric Definition 5.12, Definition 5.84
- algebraic §5.4
toric model
- of a $\mathbb{T}$ -Cartier divisor Definition 4.74
  - equivalence of Proposition-Definition 4.79
  - proper Definition 4.74
  - semipositive Definition 4.96
- of a toric variety Definition 4.56
  - morphism of Definition 4.56
toric morphism
- of toric schemes Definition 4.70
- of toric varieties Definition 4.7
toric projective bundles §8.2
- degree of §8.2
- height of §8.2
toric scheme Definition 4.55
- affine §4.5
- associated to a fan §4.5
- proper, associated to a SCR polyhedral complex §4.5
toric structure on a line bundle Definition 4.19, Remark 4.36, Definition 4.77, §4.3
toric subvariety Definition 4.12
- translated Definition 4.12
toric variety Definition 4.1
- affine §4.1
- associated to a fan §4.1
- associated to a polytope §4.4
- polarized Definition 4.41
- symplectic Remark 5.74
torus
- algebraic Example 2.8
- analytic Example 2.8
- compact Example 2.8
- over $S$ , acting on a toric scheme Definition 4.55
translate of a function §3.3
tropical Laurent polynomial Remark 3.74
V-representation
- of a piecewise affine concave function §3.5
- of a polyhedron §3.1
valuation map
- of a non-Archimedean field Definition 4.71
- on an analytic toric variety §5.1
valuation ring §2.2
- discrete §2.3
- maximal ideal of §2.2
- residue field of §2.2
variety with corners associated to a toric variety §5.1
Veronese curve §8.1
vertical curve §2.3
vertical orbit in a toric scheme §4.5
virtual support function Definition 4.15
- defining vectors Definition 4.15