9.2. Toric varieties [01CH]

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9.2. Toric varieties

We use [Ful93, KKMS73, BPS11] as references. Let $M\simeq\mathbf{Z}^{n}$ be a free abelian group, $N$ its dual, and let $T=\Spec K[M]$ be the corresponding split $K$ -torus. A projective toric $K$ -variety $X$ is described by a rational fan subdivision $\Sigma$ of $N_{\mathbf{R}}$ , and there is a natural embedding $j:N_{\mathbf{R}}\to X^{\mathrm{an}}$ given by monomial valuations that sends $n\in N_{\mathbf{R}}$ to the norm $\sum a_{m}m\in K[M]\mapsto\max\{|a_{m}|\exp(-\langle m,n\rangle)\}$ . In particular, $j(0)=x_{G}$ , the Gauss point of the open $T$ -orbit.

An ample $T$ -line bundle $L$ on $X$ defines a rational polytope $\Delta\subset M_{\mathbf{R}}$ with normal fan $\Sigma$ , such that points of $M\cap\Delta$ identify with $T$ -eigensections of $L$ .

According to [BPS11] we have the following description of toric metrics on $L$ . The polytope $\Delta$ is the Newton polytope of the piecewise $\mathbf{Q}$ -linear convex function $g_{\Delta}=\sup_{m\in\Delta}m$ on the dual space $N_{\mathbf{R}}=M_{\mathbf{R}}^{*}$ , and toric bounded (resp. model) metrics $\|\cdot\|$ on $L$ correspond to bounded (resp. piecewise $\mathbf{Q}$ -affine) functions $f$ on $N_{\mathbf{R}}$ such that $f-g_{\Delta}$ is bounded. The metric $\|\cdot\|_{f}$ attached to a function $f$ is semipositive iff $f$ is convex.

The real Monge-Ampère measure of any convex function $f$ on $N_{\mathbf{R}}$ is a well-defined positive Radon measure $\MA_{\mathbf{R}}(f)$ on $N_{\mathbf{R}}$ (see e.g. [RT77]), while the growth condition $f=g_{\Delta}+O(1)$ further guarantees that

\int_{N_{\mathbf{R}}}\MA_{\mathbf{R}}(f)=\vol(\Delta).

If $f$ is a convex function on $N_{\mathbf{R}}$ with $f=g_{\Delta}+O(1)$ , and if $\|\cdot\|_{f}$ is the corresponding continuous semipositive metric on $L$ , then [BPS11, Theorem 5.70] relates their Monge-Ampère measures as follows:

(9.2)

c_{1}(L,\|\cdot\|_{f})^{n}=n!\,j_{*}\MA_{\mathbf{R}}(f).

Since $g_{\Delta}$ is homogeneous, $\MA_{\mathbf{R}}(g_{\Delta})$ is a Dirac mass at the origin of mass $\vol(\Delta)$ , and [Ful93, p.111] implies the corresponding metric $\|\cdot\|_{g_{\Delta}}$ on $L$ to satisfy

c_{1}(L,\|\cdot\|_{g_{\Delta}})^{n}=c_{1}(L)^{n}\,\delta_{x_{G}}.

Translating in $N_{\mathbf{R}}$ we get:

Proposition 9.1.

Let $\mu$ be a Dirac mass on $X$ centered at a toric divisorial point $j(x)\in X^{\mathrm{div}}$ , $x\in N_{\mathbf{Q}}$ . Then $c_{1}(L,\|\cdot\|_{x})^{n}=\mu$ , where $\|\cdot\|_{x}$ is the toric model metric attached to the convex piecewise $\mathbf{Q}$ -affine function $y\mapsto g_{\Delta}(y-x)$ .

In the case of atomic measures supported at toric divisorial points, we can show:

Proposition 9.2.

Let $(X,L)$ be a polarized toric $K$ -variety. Pick $x_{1},...,x_{N}\in N_{\mathbf{Q}}$ and set $\mu_{w}:=\sum_{i}w_{i}\delta_{j(x_{i})}$ for each $w\in\mathbf{R}_{+}^{N}$ . Then for a dense set of $w\in\mathbf{R}_{+}^{N}\cap\{\sum_{i}w_{i}=\deg L\}$ the semipositive toric metric $\|\cdot\|$ solving

c_{1}(L,\|\cdot\|_{w})^{n}=\sum_{i}w_{i}\delta_{j(x_{i})}.

is a model metric.

Proof.

For each $t\in\mathbf{R}^{N}$ let $f_{t}$ be the upper envelope of the family of piecewise $\mathbf{Q}$ -affine convex functions $f$ on $N_{\mathbf{R}}$ such that $f=g_{\Delta}+O(1)$ and $f(x_{i})\leq t_{i}$ for all $i$ , and let $\|\cdot\|_{t}$ be the corresponding continuous toric semipositive metric. By Proposition 8.6, each measure $\mu_{w}$ with $w\in\mathbf{R}_{+}^{N}\cap\{\sum_{i}w_{i}=\deg L\}$ is of the form $c_{1}(L,\|\cdot\|_{t})^{n}$ for some $t\in\mathbf{R}^{N}$ . Now elementary Newton polytope considerations show that $f_{t}$ is piecewise $\mathbf{Q}$ -affine when all $t_{i}$ are rational, and the result follows by continuity of $t\mapsto c_{1}(L,\|\cdot\|_{t})^{n}$ . ∎

Remark 9.3.

Results of this section are likely to extend to the case of an arbitrary non-Archimedean complete non-trivially valued field. We refer to [BPR11, Gub08] for a discussion of toric varieties in this context.