Arithmetic geometry of toric varieties. Metrics, measures and heights. (English) Zbl 1311.14050

Astérisque 360. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-783-4). vi, 222 p. (2014).
A normal variety which contains an algebraic torus as an open dense subset, such that the natural action of the torus on itself extends to the whole variety, is called a toric variety. Toric varieties form an important class of algebraic varieties, whose geometry can be totally determined by combinatorial data. For instance, let \(\mathbb{T}=\mathbb{G}_m^n\) be a split torus over a field \(K\). Let \(N=\text{Hom}(\mathbb{G}_m,\mathbb{T})\simeq \mathbb{Z}^n\) 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 \(M_\mathbb{R}=M\otimes_{\mathbb{Z}}\mathbb{R}\). Then the fans \(\Sigma\) on \(N_\mathbb{R}\) correspond to the toric varieties \(X_\Sigma\) of dimension \(n\) over \(K\). \(X_\Sigma\) is proper if \(\Sigma\) is complete. A virtual support function on \(\Sigma\) is a continuous function \(\Psi: N_\mathbb{R}\to \mathbb{R}\) whose restriction to each cone of \(\Sigma\) is an element of \(M\). Such a function determines a \(\mathbb{T}\)-Cartier divisor \(D_\Psi\) and a toric line bundle \(L_\Psi=\mathcal{O}(D_\Psi)\) with a canonical toric section \(s_\Psi\) such that \(\text{div}(s_\Psi)=D_\Psi\). \(L_\Psi\) is generated by global sections if and only if \(\Psi\) is a concave function. In this case, the lattice polytope \[ \Delta_\Psi:=\{x\in M_\mathbb{R}: \langle x,u\rangle \geq \Psi(u)\text{ for all }a\in N_\mathbb{R}\}\subset M_\mathbb{R} \] conveys all the information about the pair \((X_\Sigma,L_\Psi)\), and one has the degree formula \[ \text{deg}_{L_\Psi}(X_\Sigma)=n!\text{vol}_M(\Delta_\Psi) \] where \(\text{vol}_M\) is the Haar measure on \(M_\mathbb{R}\) normalized so that \(M\) has covolume \(1\).
The content of the monograph under review is to develop an arithmetic analogue of this degree formula for heights of toric varieties. The authors have shown that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. And more generally, they established a close relation between the arithmetic geometry of toric varieties and the convex analysis. This generalizes the relation between the algebraic geometry of toric varieties and the convex geometry that we described at the beginning of this review.
To do this, the authors study the Arakelov geometry of toric varieties, including models of toric varieties over a discrete valuation ring, metrized line bundles, and their associated measures and (local/global) heights. Roughly speaking, for the global case (say \(K=\mathbb{Q}\)), given a family of concave functions \((\psi_v)_{v\in \mathcal{M}_\mathbb{Q}}\) such that \(\mid \psi_v-\Psi\mid\) is bounded for all places \(v\) and such that \(\psi_v=\Psi\) for all but a finite number of \(v\), one may endow the toric line bundle \(L:=L_\Psi\) a family of semipositive toric metrics \(\parallel\cdot\parallel_{\psi_v}\) on analytifications \((L^v)^{an}\) for all places \(v\). By definition, \(\overline{L}=(L,(\parallel\cdot\parallel_{\psi_v})_v)\) becomes a semipositive quasi-algebraic metrized toric line bundle. Moreover, every semipositive quasi-algebraic toric metric on \(L\) arises in this way. To all places \(v\), the associated roof functions \(\vartheta_{\overline{L}^v,s}: \Delta_\Psi\to \mathbb{R}\) are zero except for a finite number of places. Then, the global height of \(X\) with respect to \(\overline{L}\) can be computed as \[ h_{\overline{L}}(X)=\sum_{v\in \mathcal{M}_\mathbb{Q}}h_{\overline{L}^v}^{\text{tor}}(X_v)=(n+1)!\sum_{v\in \mathcal{M}_\mathbb{Q}}\int_{\Delta_\Psi}\vartheta_{\overline{L}^v,s}d\text{vol}_M. \]
Apart from this formula, the authors also presented a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This is helpful in computing the height of toric varieties with respect to some interesting metrics arising from polytopes. As applications, they computed the height of toric projective curves with respect to the Fubini-Study metric, and of some toric bundles.


14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
52A41 Convex functions and convex programs in convex geometry
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