Systems of polynomial equations appear in a wide variety of contexts in both pure and applied mathematics. Systems arising from applications are not random but come with a certain structure. When studying those systems, it is important to be able to exploit that structure.

A relevant result in this direction is the Bernštein-Kušnirenko-Khovanskii theorem [Kuš76, Ber75]. Let $K$ be a field with algebraic closure ${\overline{K}}$. Let $\Delta\subset\mathbb{R}^{n}$ be a lattice polytope and $f_{1},\dots,f_{n}\in K[t_{1}^{\pm 1},\dots,t_{n}^{\pm 1}]$ a family of Laurent polynomials whose Newton polytope is contained in $\Delta$. The BKK theorem implies that the number (counting multiplicities) of isolated common zeros of $f_{1},\dots,f_{n}$ in $({\overline{K}}^{\times})^{n}$ is bounded above by $n!$ times the volume of $\Delta$, with equality when $f_{1},\dots,f_{n}$ is generic among the families of Laurent polynomials with Newton polytope contained in $\Delta$. This shows how a geometric problem (the counting of the number of solutions of a system of equations) can be translated into a combinatorial, simpler one. It is commonly used to predict when a given system of polynomial equations has a small number of solutions. As such, it is a cornerstone of polynomial equation solving and has motivated a large amount of work and results over the past 25 years, see for instance [GKZ94, Stu02, PS08b] and the references therein.

A natural way to study polynomials with prescribed Newton polytope is to associate to the polytope $\Delta$ a toric variety $X$ over $K$ equipped with an ample line bundle $L$. The polytope conveys all the information about the pair $(X,L)$. For instance, the degree of $X$ with respect to $L$ is given by the formula

(1.1)

$$\deg_{L}(X)=n!\operatorname{vol}(\Delta),$$

where $\operatorname{vol}$ denotes the Lebesgue measure of $\mathbb{R}^{n}$. The Laurent polynomials $f_{i}$ can be identified with global sections of $L$, and the BKK theorem is equivalent to this formula. Indeed, there is a dictionary which allows to translate algebro-geometric properties of toric varieties in terms of combinatorial properties of polytopes and fans, and formula (1.1) is one entry in this “toric dictionary”.

The central motivation for this text is an arithmetic analogue for heights of this formula, which is Theorem 1.2 below. The height is a basic arithmetic invariant of a proper variety over the field of rational numbers. Together with its degree, it measures the amount of information needed to represent this variety, for instance, via its Chow form. Hence, this invariant is also relevant in computational algebraic geometry, see for instance [GHH+97, AKS07, DKS10]. The notion of height of varieties generalizes the height of points already considered by Siegel, Northcott, Weil and others, and is a key tool in Diophantine geometry, see for instance [BG06] and the references therein.

Assume that the pair $(X,L)$ is defined over $\mathbb{Q}$. Let $\mathfrak{M}_{\mathbb{Q}}$ denote the set of places of $\mathbb{Q}$, and $\{\vartheta_{v}\}_{v\in\mathfrak{M}_{\mathbb{Q}}}$ a family of concave functions on $\Delta$ such that $\vartheta_{v}\equiv 0$ for all but a finite number of $v$. We will show that, to this data, one can associate an adelic family of metrics $\{\|\cdot\|_{v}\}_{v}$ on $L$. Write ${\overline{L}}=(L,\{\|\cdot\|_{v}\}_{v})$ for the resulting metrized line bundle.

This theorem was announced in [BPS09] and we prove it in the present text. To establish it in a wide generality, we have been led to study the Arakelov geometry of toric varieties. In the course of our research, we have found that a large part of the arithmetic geometry of toric varieties can be translated in terms of convex analysis. In particular, we have added a number of new entries to the arithmetic geometry chapter of the toric dictionary, including models of toric varieties over a discrete valuation ring, metrized line bundles, and their associated measures and heights. These objects are closely related to objects of convex analysis like polyhedral complexes, concave functions, Monge-Ampère measures and Legendre-Fenchel duality.

These additions to the toric dictionary are very concrete and well-suited for computations. In particular, they provide a new wealth of examples in Arakelov geometry where constructions can be made explicit and properties tested. In this direction, we also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This formula allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. Some of these heights are related to the average entropy of a simple random process on the polytope. We also compute the height of toric projective curves with respect to the Fubini-Study metric and of some toric bundles.

There are many other arithmetic invariants of toric varieties that may be studied in terms of convex analysis. For instance, one can give criteria for positivity properties of toric metrized line bundles, like having or being generated by small sections, a formula for its arithmetic volume, and an arithmetic analogue of the BKK theorem bounding the height of the solutions of a system of sparse polynomial equations with rational coefficients. In fact, we expect that the results of this text are just the starting point of a program relating the arithmetic geometry of toric varieties and convex analysis.

In the rest of this introduction, we will present the context and the contents of our results. We will refer to the body of the text for the precise definitions and statements.

Arakelov geometry provides a framework to define and study heights. We leave for a moment the realm of toric varieties, and we consider a projective variety $X$ over $\mathbb{Q}$ of dimension $n$. Let ${\mathcal{X}}$ be a proper integral model of $X$, and $X(\mathbb{C})$ the analytic space over the complex numbers associated to $X$. The main idea behind Arakelov geometry is that the pair $({\mathcal{X}},X(\mathbb{C}))$ should behave like a compact variety of dimension $n+1$ [Ara74]. Following this philosophy, Gillet and Soulé have developed an arithmetic intersection theory [GS90]. As an application of this theory, one can introduce a very general and precise definition, with a geometric flavor, of the height of a variety [BGS94]. To the variety $X$, one associates the arithmetic intersection ring $\operatorname{\widehat{CH}}^{*}(X)_{\mathbb{Q}}$. This ring is equipped with a trace map $\int\colon\operatorname{\widehat{CH}}^{n+1}(X)_{\mathbb{Q}}\to\mathbb{R}$. Given a line bundle $L$ on $X$, an arithmetic line bundle ${\overline{L}}$ is a pair $({\mathcal{L}},\|\cdot\|)$, where ${\mathcal{L}}$ is a line bundle on ${\mathcal{X}}$ which is an integral model of $L$, and $\|\cdot\|$ is a smooth metric on the analytification of $L$. In this setting, the analogue of the first Chern class of $L$ is the arithmetic first Chern class $\operatorname{\widehat{c}_{1}}({\overline{L}})\in\operatorname{\widehat{CH}}^{1}(X)_{\mathbb{Q}}$. The height of $X$ with respect to ${\overline{L}}$ is then defined as

$$\operatorname{h}_{{\overline{L}}}(X)=\int\operatorname{\widehat{c}_{1}}({\overline{L}})^{n+1}\in\mathbb{R}.$$

This is the arithmetic analogue of the degree of $X$ with respect to $L$. This formalism has allowed to obtain arithmetic analogues of important results in algebraic geometry like the Bézout’s theorem, the Riemman-Roch theorem, the Lefschetz fixed point formula, the Hilbert-Samuel formula, etc.

This approach has two technical issues. In the first place, it only works for smooth varieties and smooth metrics. In the second place, it depends on the existence of an integral model, which puts the Archimedean and non-Archimedean places in different footing. For the definition of heights, both issues were addressed by Zhang [Zha95b] by taking an adelic point of view and considering uniform limits of semipositive metrics.

Many natural metrics that arise when studying line bundles on toric varieties are not smooth, but are particular cases of the metrics considered by Zhang. This is the case for the canonical metric of a toric line bundle, see §5.2. The associated canonical height of subvarieties plays an important role in Diophantine approximation in tori, in particular in the generalized Bogomolov and Lehmer problems, see for instance [DP99, AV09] and the references therein. Maillot has extended the arithmetic intersection theory of Gillet and Soulé to this kind of metrics at the Archimedean place, while maintaining the use of an integral model to handle the non-Archimedean places [Mai00].

The adelic point of view of Zhang was developed by Gubler [Gub02, Gub03] and by Chambert-Loir [Cha06]. From this point of view, the height is defined as a sum of local contributions. We outline this procedure, that will be recalled with more detail in §2.

For the local case, let $K$ be either $\mathbb{R}$, $\mathbb{C}$, or a field complete with respect to a nontrivial non-Archimedean absolute value. Let $X$ be a proper variety over $K$ and $L$ a line bundle on $X$, and consider their analytifications, respectively denoted by $X^{{\text{\rm an}}}$ and $L^{{\text{\rm an}}}$. In the Archimedean case, $X^{\text{\rm an}}$ is the complex space $X(\mathbb{C})$ (equipped with an anti-linear involution, if $K=\mathbb{R}$), whereas in the non-Archimedean case it is the Berkovich space associated to $X$. The basic metrics that can be put on $L^{\text{\rm an}}$ are the smooth metrics in the Archimedean case, and the algebraic metrics in the non-Archimedean case, that is, the metrics induced by an integral model of a pair $(X,L^{\otimes e})$ with $e\geq 1$. There is a notion of semipositivity for smooth and and for algebraic metrics, and the uniform limit of such semipositive metrics leads to the notion of approachable metric on $L^{{\text{\rm an}}}$. More generally, a metric on $L^{{\text{\rm an}}}$ is integrable if it is the quotient of two approachable metrics.

Let ${\overline{L}}$ be an integrable metrized line bundle on $X$ and $Y$ a $d$-dimensional cycle of $X$. These data induce a (signed) measure on $X^{{\text{\rm an}}}$, denoted $c_{1}({\overline{L}})^{\wedge d}\wedge\delta_{Y}$ by analogy with the Archimedean smooth case, where it corresponds with the current of integration along $Y^{{\text{\rm an}}}$ of the $d$-th power of the first Chern form. This measure plays an important role in the distribution of points of small height in the direction of the Bogomolov conjecture and its generalizations, see for instance [Bil97, SUZ97, Yua08]. Furthermore, if we have sections $s_{i}$, $i=0,\dots,d$, that meet $Y$ properly, one can define a notion of local height $\operatorname{h}_{{\overline{L}}}(Y;s_{0},\dots,s_{d})$. The metrics and their associated measures and local heights are related by the Bézout-type formula:

$$\operatorname{h}_{{\overline{L}}}(Y\cdot\operatorname{div}(s_{d});s_{0},\dots,s_{d-1})=\operatorname{h}_{{\overline{L}}}(Y;s_{0},\dots,s_{d})+\int_{X^{{\text{\rm an}}}}\log||s_{d}||\,c_{1}({\overline{L}})^{\wedge d}\wedge\delta_{Y}.$$

For the global case, consider a proper variety $X$ over $\mathbb{Q}$ and a line bundle $L$ on $X$. For simplicity, assume that $X$ is projective, although this hypothesis is not really necessary. An integrable quasi-algebraic metric on $L$ is a family of integrable metrics $\|\cdot\|_{v}$ on the analytic line bundles $L_{v}^{{\text{\rm an}}}$, $v\in\mathfrak{M}_{\mathbb{Q}}$, such that there is an integral model of $(X,L^{\otimes e})$, $e\geq 1$, which induces $\|\cdot\|_{v}$ for all but a finite number of $v$. Write ${\overline{L}}=(L,\{\|\cdot\|_{v}\}_{v})$, and ${\overline{L}}_{v}=(L_{v},\|\cdot\|_{v})$ for each $v\in\mathfrak{M}_{\mathbb{Q}}$. Given a $d$-dimensional cycle $Y$ of $X$, its global height is defined as

$$\operatorname{h}_{{\overline{L}}}(Y)=\sum_{v\in\mathfrak{M}_{\mathbb{Q}}}\operatorname{h}_{{\overline{L}}_{v}}(Y;s_{0},\dots,s_{d}),$$

for any family of sections $s_{i}$, $i=0,\dots,d$, meeting $Y$ properly. The fact that the metric is quasi-algebraic implies that the right-hand side has only a finite number of nonzero terms, and the product formula implies that this definition does not depend on the choice of sections. This notion can be extended to number fields, function fields and, more generally, to $M$-fields [Zha95b, Gub03].

Now we review briefly the elements of the construction of toric varieties from combinatorial data, see §4 for more details. Let $K$ be a field and $\mathbb{T}\simeq\mathbb{G}_{m}^{n}$ a split torus over $K$. Let $N=\operatorname{Hom}(\mathbb{G}_{m},\mathbb{T})\simeq\mathbb{Z}^{n}$ be the lattice of one-parameter subgroups of $\mathbb{T}$ and $M=N^{\vee}$ the dual lattice of characters of $\mathbb{T}$. Set $N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}{\mathbb{R}}$ and $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$. To a fan $\Sigma$ on $N_{\mathbb{R}}$ one can associate a toric variety $X_{\Sigma}$ of dimension $n$. It is a normal variety that contains $\mathbb{T}$ as a dense open subset, denoted $X_{\Sigma,0}$, and there is an action of $\mathbb{T}$ on $X_{\Sigma}$ which extends the natural action of the torus on itself. In particular, every toric variety has a distinguished point $x_{0}$ that corresponds to the identity element of $\mathbb{T}$. The variety $X_{\Sigma}$ is proper whenever the underlying fan is complete. For sake of simplicity, in this introduction we will restrict to the proper case.

A Cartier divisor invariant under the torus action is called a $\mathbb{T}$-Cartier divisor. In combinatorial terms, a $\mathbb{T}$-Cartier divisor is determined by a virtual support function on $\Sigma$, that is, a continuous function $\Psi\colon N_{\mathbb{R}}\rightarrow{\mathbb{R}}$ whose restriction to each cone of $\Sigma$ is an element of $M$. Let $D_{\Psi}$ denote the $\mathbb{T}$-Cartier divisor of $X_{\Sigma}$ determined by $\Psi$. A toric line bundle on $X_{\Sigma}$ is a line bundle $L$ on this toric variety, together with the choice of a nonzero element $z\in L_{x_{0}}$. The total space of a toric line bundle has a natural structure of toric variety whose distinguished point agrees with $z$. A rational section of a toric line bundle is called toric if it is regular and nowhere zero on the principal open subset $X_{\Sigma,0}$, and $s(x_{0})=z$. Given a virtual support function $\Psi$, the line bundle $L_{\Psi}={\mathcal{O}}(D_{\Psi})$ has a natural structure of toric line bundle and a canonical toric section $s_{\Psi}$ such that $\operatorname{div}(s_{\Psi})=D_{\Psi}$. Indeed, any line bundle on $X_{\Sigma}$ is isomorphic to a toric line bundle of the form $L_{\Psi}$ for some $\Psi$. The line bundle $L_{\Psi}$ is generated by global sections (respectively, is ample) if and only if $\Psi$ is concave (respectively, $\Psi$ is strictly concave on $\Sigma$).

Consider the lattice polytope

$$\Delta_{\Psi}:=\{x\in M_{\mathbb{R}}:\langle x,u\rangle\geq\Psi(u)\mbox{ for all }u\in N_{\mathbb{R}}\}\subset M_{\mathbb{R}}.$$

This polytope encodes a lot of information about the pair $(X_{\Sigma},L_{\Psi})$. In case the virtual support function $\Psi$ is concave, it is determined by this polytope, and the formula (1.1) can be written more precisely as

$$\deg_{L_{\Psi}}(X_{\Sigma})=n!\operatorname{vol}_{M}(\Delta_{\Psi}),$$

where the volume is computed with respect to the Haar measure $\operatorname{vol}_{M}$ on $M_{\mathbb{R}}$ normalized so that $M$ has covolume 1.

In this text we extend the toric dictionary to metrics, measures and heights as considered above. For the local case, let $K$ be either $\mathbb{R}$, $\mathbb{C}$, or a field complete with respect to a nontrivial non-Archimedean absolute value associated to a discrete valuation. In this latter case, let $K^{\circ}$ be the valuation ring, $K^{\circ\circ}$ its maximal ideal and $\varpi$ a generator of $K^{\circ\circ}$. Let $\mathbb{T}$ be an $n$-dimensional split torus over $K$, $\mathbb{T}^{{\text{\rm an}}}$ its analytification and $\mathbb{S}^{{\text{\rm an}}}$ the compact torus of $\mathbb{T}^{{\text{\rm an}}}$. Let $X$ be a toric variety over $K$ with torus $\mathbb{T}$ and $L$ a toric line bundle on $X$. The compact torus $\mathbb{S}^{{\text{\rm an}}}$ is a closed subgroup of the analytic torus $\mathbb{T}^{{\text{\rm an}}}$ and it acts on $X^{{\text{\rm an}}}$. A metric $\|\cdot\|$ on $L^{{\text{\rm an}}}$ is toric if, for every toric section $s$, the function $\|s\|$ is invariant under the action of $\mathbb{S}^{{\text{\rm an}}}$.

The correspondence that to a virtual support function assigns a toric line bundle with a toric section can be extended to approachable and integrable metrics. Assume that $\Psi$ is concave, and let $X_{\Sigma}$, $L_{\Psi}$ and $s_{\Psi}$ be as before. For short, write $X=X_{\Sigma}$, $L=L_{\Psi}$ and $s=s_{\Psi}$. There is a fibration ${\operatorname{val}}_{K}\colon X_{0}^{{\text{\rm an}}}\to N_{\mathbb{R}}$ whose fibers are the orbits of the action of $\mathbb{S}^{{\text{\rm an}}}$ on $X_{0}^{{\text{\rm an}}}$. Now let $\psi\colon N_{\mathbb{R}}\to\mathbb{R}$ be a continuous function. We define a metric on the restriction $L^{{\text{\rm an}}}|_{X_{0}^{{\text{\rm an}}}}$ by setting

$$\|s(p)\|_{\psi}=e^{\lambda_{K}\psi({\operatorname{val}}_{K}(p))},$$

with $\lambda_{K}=1$ if $K=\mathbb{R}$ or $\mathbb{C}$, and $\lambda_{K}=-\log|\varpi|$ otherwise.

Our first addition to the toric dictionary is the following classification result. Assume that the function $\psi$ is concave and that $|\psi-\Psi|$ is bounded. Then $\|\cdot\|_{\psi}$ extends to an approachable toric metric on $L^{{\text{\rm an}}}$ and, moreover, every approachable toric metric on $L^{{\text{\rm an}}}$ arises in this way (Theorem 5.73(1)). There is a similar characterization of integrable toric metrics in terms of differences of concave functions (Corollary 5.83) and a characterization of toric metrics that involves the topology of the variety with corners associated to $X_{\Sigma}$ (Proposition 5.16). As a consequence of these classification results, we obtain a new interpretation of the canonical metric of $L^{{\text{\rm an}}}$ as the metric associated to the concave function $\Psi$ under this correspondence.

We can also classify approachable metrics in terms of concave functions on polytopes: there is a bijective correspondence between the space of continuous concave functions on $\Delta_{\Psi}$ and the space of approachable toric metrics on $L^{{\text{\rm an}}}$ (Theorem 5.73(2)). This correspondence is induced by the previous one and the Legendre-Fenchel duality of concave functions. Namely, let $\|\cdot\|$ be an approachable toric metric on $L^{{\text{\rm an}}}$, write ${\overline{L}}=(L,\|\cdot\|)$ and let $\psi$ be the corresponding concave function. The associate roof function $\vartheta_{{\overline{L}},s}\colon\Delta_{\Psi}\to\mathbb{R}$ is the concave function defined as $\lambda_{K}$ times the Legendre-Fenchel dual $\psi^{\vee}$. One of the main outcomes of this text is that the pair $(\Delta_{\Psi},\vartheta_{{\overline{L}},s})$ plays, in the arithmetic geometry of toric varieties, a role analogous to that of the polytope in its algebraic geometry.

Our second addition to the dictionary is the following characterization of the measure associated to an approachable toric metric. Let $X$, ${\overline{L}}$ and $\psi$ be as before, and write $\mu_{\psi}=c_{1}^{\wedge n+1}({\overline{L}})\wedge\delta_{X_{\Sigma}}$ for the induced measure on $X^{{\text{\rm an}}}$. Then

$$({\operatorname{val}}_{K})_{*}(\mu_{\psi}|_{X^{{\text{\rm an}}}_{0}})=n!\,{\mathcal{M}}_{M}(\psi),$$

where ${\mathcal{M}}_{M}(\psi)$ is the (real) Monge-Ampère measure of $\psi$ with respect to the lattice $M$ (Definition 3.92). The measure $\mu_{\psi}$ is determined by this formula, and the conditions of being invariant under the action of $\mathbb{S}^{{\text{\rm an}}}$ and that the set $X^{{\text{\rm an}}}\setminus X_{0}^{{\text{\rm an}}}$ has measure zero. This gives a direct and fairly explicit expression for the measure associated to an approachable toric metric.

The fact that each toric line bundle has a canonical metric allows us to introduce a notion of local toric height that is independent of a choice of sections. Let $X$ be an $n$-dimensional projective toric variety and ${\overline{L}}$ an approachable toric line bundle as before, and let ${\overline{L}}^{{\operatorname{can}}}$ be the same toric line bundle $L$ equipped with the canonical metric. The toric local height of $X$ with respect to ${\overline{L}}$ is defined as

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X)=\operatorname{h}_{{\overline{L}}}(X;s_{0},\dots,s_{n})-\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(X;s_{0},\dots,s_{n}),$$

for any family of sections $s_{i}$, $i=0,\dots,d$, that meet properly on $X$ (Definition 6.1). Our third addition to the toric dictionary is the following formula for this toric local height in terms of the roof function introduced above (Theorem 6.6):

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X)=(n+1)!\int_{\Delta_{\Psi}}\vartheta_{{\overline{L}},s}\,\text{\rm d}\operatorname{vol}_{M}.$$

More generally, the toric local height can be defined for a family of $n+1$ integrable toric line bundles on $X$. The formula above can be extended by multilinearity to compute this local toric height in terms of the mixed integral of the associated roof functions (Remark 6.23).

For the global case, let $\Sigma$ and $\Psi$ be as before, and consider the associated toric variety $X$ over $\mathbb{Q}$ equipped with a toric line bundle $L$ and toric section $s$. Given a family of concave functions $\{\psi_{v}\}_{v\in\mathfrak{M}_{\mathbb{Q}}}$ such that $|\psi_{v}-\Psi|$ is bounded for all $v$ and such that $\psi_{v}=\Psi$ for all but a finite number of $v$, the metrized toric line bundle ${\overline{L}}=(L,\{\|\cdot\|_{\psi_{v}}\}_{v})$ is quasi-algebraic. Moreover, every approachable quasi-algebraic toric metric on $L$ arises in this way (Theorem 5.85). Write ${\overline{L}}^{v}=(L,\|\cdot\|_{\psi_{v}})$ for the metrized toric line bundle corresponding to a place $v$. The associated roof functions $\vartheta_{{\overline{L}}^{v},s}\colon\Delta_{\Psi}\to\mathbb{R}$ are identically zero except for a finite number of places. Then, the global height of $X$ with respect to ${\overline{L}}$ can be computed as (Theorem 6.37)

$$\operatorname{h}_{{\overline{L}}}(X)=\sum_{v\in\mathfrak{M}_{\mathbb{Q}}}\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}^{v}}(X_{v})=(n+1)!\sum_{v\in\mathfrak{M}_{\mathbb{Q}}}\int_{\Delta_{\Psi}}\vartheta_{{\overline{L}}^{v},s}\,\text{\rm d}\operatorname{vol}_{M},$$

which precises Theorem 1.2 at the beginning of this introduction.

A remarkable feature of these results is that they read exactly the same in the Archimedean and in the non-Archimedean cases. For general metrized line bundles, these two cases are analogous but not identical. By contrast, the classification of toric metrics and the formulae for the associated measures and for the local heights are the same in both cases. We also point out that these results holds in greater generality than explained in this introduction: in particular, they hold for proper toric varieties which are not necessarily projective and, in the global case, for general adelic fields (Definition 2.47). By contrast, we content ourselves with the case when the torus is split. For the computation of heights, one can always reduce to the split case by considering a suitable field extension. Still, it would be interesting to extend our results to the non-split case by considering the corresponding Galois actions as, for instance, in [ELST10].

The toric dictionary in arithmetic geometry is very concrete and well-suited for computations. For instance, let $K$ be a local field, $X$ a toric variety and $\varphi\colon X\to\mathbb{P}^{r}$ an equivariant map. Let ${\overline{L}}$ be the toric approachable metrized line bundle on $X$ induced by the canonical metric on the universal line bundle of $\mathbb{P}^{r}$, and $s$ a toric section of $L$. The concave function $\psi\colon N_{\mathbb{R}}\to\mathbb{R}$ corresponding to this metric is piecewise affine (Example 5.26). Hence, it defines a polyhedral complex in $N_{\mathbb{R}}$, and it turns out that $({\operatorname{val}}_{K})_{*}(\mu_{\psi}|_{X_{0}^{{\text{\rm an}}}})$, the direct image under ${\operatorname{val}}_{K}$ of the measure induced by ${\overline{L}}$, is a discrete measure on $N_{\mathbb{R}}$ supported on the vertices of this polyhedral complex (Proposition 3.95). The roof function $\vartheta_{{\overline{L}},s}$ is the function parameterizing the upper envelope of a polytope in $M_{\mathbb{R}}\times\mathbb{R}$ associated to $\varphi$ and the section $s$ (Example 6.31). The toric local height of $X$ with respect to ${\overline{L}}$ can be computed as the integral of this piecewise affine concave function.

Another nice example is given by toric bundles on a projective space. For a finite sequence of integers $a_{r}\geq\dots\geq a_{0}\geq 1$, we consider the vector bundle on $\mathbb{P}^{n}_{\mathbb{Q}}$

$$E:=\mathcal{O}(a_{0})\oplus\dots\oplus\mathcal{O}(a_{r}).$$

The toric bundle $\mathbb{P}(E)\rightarrow\mathbb{P}^{n}_{\mathbb{Q}}$ is defined as the bundle of hyperplanes of the total space of $E$. This is an $(n+r)$-dimensional toric variety over $\mathbb{Q}$ which can be equipped with an ample universal line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$, see §8.2 for details.

We equip ${\mathcal{O}_{\mathbb{P}(E)}(1)}$ with an approachable adelic toric metric as follows: the Fubini-Study metrics on each line bundle $\mathcal{O}(a_{j})$ induces a semipositive smooth toric metric on $\mathcal{O}_{\mathbb{P}(E)}(1)$ for the Archimedean place of $\mathbb{Q}$, whereas for the finite places we consider the corresponding canonical metric. We show that both the corresponding concave functions $\psi_{v}$ and roof functions $\vartheta_{v}$ can be described in explicit terms (Lemma 8.17 and Proposition 8.20). We can then compute the height of $\mathbb{P}(E)$ with respect to this metrized line bundle as (Proposition 8.26)

$$\operatorname{h}_{{\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}}}(\mathbb{P}(E))=\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})\hskip-4.2679pt\sum_{{\boldsymbol{i}\in\mathbb{N}^{r+1}}\atop{|\boldsymbol{i}|=n+1}}\boldsymbol{a}^{\boldsymbol{i}}+\sum_{{\boldsymbol{i}\in\mathbb{N}^{r+1}}\atop{|\boldsymbol{i}|=n}}A_{n,r}(\boldsymbol{i})\,\boldsymbol{a}^{\boldsymbol{i}},$$

where for $\boldsymbol{i}=(i_{0},\dots,i_{r})\in\mathbb{N}^{r+1}$, we set $|\boldsymbol{i}|=i_{0}+\dots+i_{r}$, $\boldsymbol{a}^{\boldsymbol{i}}=a_{0}^{i_{0}}\dots a_{r}^{i_{r}}$ and $A_{n,r}(\boldsymbol{i})=\sum_{m=0}^{r}(i_{m}+1)\sum_{j=i_{m}+2}^{n+r+1}\frac{1}{2j}$, while $\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n}_{\mathbb{Q}})=\sum_{i=1}^{n}\sum_{j=1}^{i}\frac{1}{2j}$ denotes the height of the projective space with respect to the Fubini-Study metric. In particular, the height of $\mathbb{P}(E)$ is a positive rational number.

The Fubini-Study height of the projective space was computed by Bost, Gillet and Soulé [BGS94, Lemma 3.3.1]. Other early computations for the Fubini-Study height of some toric hypersurfaces where obtained in [CM00, Dan97]. Mourougane has determined the height of Hirzebruch surfaces, as a consequence of his computations of Bott-Chern secondary classes [Mou06]. A Hirzebruch surface is a toric bundle over $\mathbb{P}^{1}_{\mathbb{Q}}$, and the result of Mourougane is a particular case of our computations for the height of toric bundles, see Remark 8.27.

The fact that the canonical height of a toric variety is zero is well-known. It results from its original construction by a limit process on the direct images of the variety under the so-called “powers maps”. Maillot has studied the Arakelov geometry of toric varieties and line bundles with respect to the canonical metric, including the associated Chern currents and their product [Mai00].

In [PS08a], Philippon and Sombra gave a formula for the canonical height of a “translated” toric projective variety, a projective variety which is the closure of a translate of a subtorus, defined over a number field. In [PS08b], they also obtain a similar formula for the function field case. Both results are particular cases of our general formula, see Remark 6.40. Indeed, part of our motivation for the present text was to understand and generalize this formula in the framework of Arakelov geometry.

For the Archimedean smooth case, our constructions are related to the Guillemin-Abreu classification of Kähler structures on symplectic toric varieties [Abr03]. The roof function corresponding to a smooth metrized line bundle on a smooth toric variety coincides, up to a sign, with the so-called “symplectic potential” of a Kähler toric variety, see Remark 5.74. In the Archimedean continuous case, Boucksom and Chen have recently considered a similar construction in their study of arithmetic Okounkov bodies [BC09]. It would be interesting to further explore the connection with these results.

We now discuss the contents of each section, including some other results of interest.

Section 2 is devoted to the first half of the dictionary. Namely, we review integrable metrized line bundles both in the Archimedean and in the non-Archimedean cases. For the latter case, we recall the basic properties of Berkovich spaces of schemes. We then explain the associated measures and heights following [Zha95b, Cha06, Gub03]. For simplicity, the theory presented is not as general as the one in [Gub03]: in the non-Archimedean case we restrict ourselves to discrete valuation rings and in the global case to adelic fields, while in loc. cit. the theory is developed for arbitrary valuations and for $M$-fields, respectively.

Section 3 deals with the second half of the dictionary, that is, convex analysis with emphasis on polyhedral sets. Most of the material in this section is classical. We have gathered all the required results, adapting them to our needs and adding some new ones. We work with concave functions, which are the functions which naturally arise in the theory of toric varieties. For latter reference, we have translated many of the notions and results of convex analysis, usually stated for convex functions, in terms of concave functions.

We first recall the basic definitions about convex sets and convex decompositions, and then we study concave functions and the Legendre-Fenchel duality. We introduce a notion of Legendre-Fenchel correspondence for general closed concave functions, as a duality between convex decompositions (Definition 3.31 and Theorem 3.33). This is the right generalization of both the classical Legendre transform of strictly concave differentiable functions, and the duality between polyhedral complexes induced by a piecewise affine concave function. We also consider the interplay between Legendre-Fenchel duality and operations on concave functions like, for instance, the direct and inverse images by affine maps. This latter study will be important when considering the functoriality with respect to equivariant morphisms between toric varieties. We next particularize to two extreme cases: differentiable concave functions whose stability set is a polytope that will be related to semipositive smooth toric metrics in the Archimedean case, and to piecewise affine concave functions that will correspond to semipositive algebraic toric metrics in the non-Archimedean case. Next, we treat differences of concave functions, that will be related to integrable metrics. We end this section by studying the Monge-Ampère measure associated to a concave function. There is an interesting interplay between Monge-Ampère measures and Legendre-Fenchel duality. In this direction, we prove a combinatorial analogue of the arithmetic Bézout’s theorem (Theorem 3.97), which is a key ingredient in the proof of our formulae for the height of a toric variety.

In §4 we study the algebraic geometry of toric varieties over a field and of toric schemes over a discrete valuation ring (DVR). We start by recalling the basic constructions and results on toric varieties, including Cartier and Weil divisors, toric line bundles and sections, orbits and equivariant morphisms, and positivity properties. Toric schemes over a DVR where first considered by Mumford in [KKMS73], who studied and classified them in terms of fans in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$. In the proper case, these schemes can be alternatively classified in terms of complete polyhedral complexes in $N_{\mathbb{R}}$ [BS10]. Given a complete fan $\Sigma$ in $N_{\mathbb{R}}$, the models over a DVR of the proper toric variety $X_{\Sigma}$ are classified by complete polyhedral complexes on $N_{\mathbb{R}}$ whose recession fan (Definition 3.7) coincides with $\Sigma$ (Theorem 4.60). Let $\Pi$ be such a polyhedral complex, and denote by ${\mathcal{X}}_{\Pi}$ the corresponding model of $X_{\Sigma}$. Let $(L,s)$ be a toric line bundle on $X_{\Sigma}$ with a toric section defined by a virtual support function $\Psi$. We show that the models of $(L,s)$ over ${\mathcal{X}}_{\Pi}$ are classified by functions that are rational piecewise affine on $\Pi$ and whose recession function is $\Psi$ (Theorem 4.81). We also prove a toric version of the Nakai-Moishezon criterion for toric schemes over a DVR, which implies that semipositive models translate into concave functions under the above correspondence (Theorem 4.95).

In §5 we study toric metrics and their associated measures. For the discussion, consider a local field $K$, a complete fan $\Sigma$ on $N_{\mathbb{R}}$ and a virtual support function $\Psi$ on $\Sigma$, and let $(X,L)$ denote the corresponding proper toric variety over $K$ and toric line bundle. We first introduce a variety with corners $N_{\Sigma}$ which is a compactification of $N_{\mathbb{R}}$, together with a proper map ${\operatorname{val}}_{K}\colon X_{\Sigma}^{{\text{\rm an}}}\to N_{\Sigma}$ whose fibers are the orbits of the action of $\mathbb{S}^{{\text{\rm an}}}$ on $X_{\Sigma}^{{\text{\rm an}}}$, and we prove the classification theorem for toric metrics on $L^{{\text{\rm an}}}$ (Proposition 5.16). We next treat smooth metrics in the Archimedean case. A toric smooth metric is semipositive if and only if the associated function $\psi$ is concave (Proposition 5.29). We make explicit the associated measure in terms of the Hessian of this function, hence in terms of the Monge-Ampère measure of $\psi$ (Theorem 5.33). We also observe that an arbitrary smooth metric on $L$ can be turned into a toric smooth metric by averaging it by the action of $\mathbb{S}^{{\text{\rm an}}}$. If the given metric is semipositive, so is the obtained toric smooth metric.

Next, in the same section, we consider algebraic metrics in the non-Archimedean case. We first show how to describe the reduction map for toric schemes over a DVR in terms of the corresponding polyhedral complex and the map ${\operatorname{val}}_{K}$ (Lemma 5.39). We then study the triangle formed by toric metrics, rational piecewise affine functions and toric models (Proposition 5.41 and Theorem 5.49), the problem of obtaining a toric metric from a non-toric one (Proposition 5.51) and the effect of a field extension (Proposition 5.53). Next, we treat in detail the one-dimensional case, were one can write in explicit terms the metrics, associated functions and measures. Back to the general case, we use these results to complete the characterization of toric semipositive algebraic metrics in terms of piecewise affine concave functions (Proposition 5.67). We also describe the measure associated to a semipositive toric algebraic metric in terms of the Monge-Ampère measure of its associated concave function (Theorem 5.70).

Once we have studied smooth metrics in the Archimedean case and algebraic metrics in the non-Archimedean case, we can study approachable toric metrics. We show that the same classification theorem is valid in the Archimedean and non-Archimedean cases (Theorem 5.73). Moreover, the associated measure is described in exactly the same way in both cases (Theorem 5.81). We end this section by introducing and classifying adelic toric metrics (Definition 5.84 and Corollary 5.83).

In §6, we prove the formulae for the toric local height and for the global height of toric varieties (Theorem 6.6 and Theorem 6.37). By using the functorial properties of the height, we recover, from our general formula, the formulae for the canonical height of a translated toric projective variety in [PS08a, Théorème 0.3] for number fields and in [PS08b, Proposition 4.1] for function fields.

In §7, we consider the problem of integrating functions on polytopes. We first present a closed formula for the integral over a polytope of a function of one variable composed with a linear form, extending in this direction Brion’s formula for the case of a simplex [Bri88] (Proposition 7.3 and Corollary 7.14). This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes (Proposition 7.27). We can interpret some of these heights as the average entropy of a simple random process defined by the polytope (Proposition 7.34).

In §8 we study some further examples. We first consider translated toric curves in $\mathbb{P}^{n}_{\mathbb{Q}}$. For these curves, we consider the line bundle obtained from the restriction of ${\mathcal{O}}(1)$ to the curve, equipped with the metric induced by the Fubiny-Study metric at the place at infinity and by the canonical metric for the finite places. We compute the corresponding concave function $\psi$ and toric local height in terms of the roots of a univariate polynomial (Theorem 8.7). We finally consider toric bundles as explained before, and compute the relevant concave functions, measure and height.

Part of this work was done while the authors met at the Universitat de Barcelona, the Centre de Recerca Matemàtica (Barcelona), the Institut de Mathématiques de Jussieu (Paris), and the Université de Bordeaux 1. Short courses on parts of this text were delivered at the Morningside Center of Mathematics (Beijing), the Centro de Investigación en Matemáticas (Guanajuato), and the Universidad de Buenos Aires. We thank all of these institutions for their hospitality.