5.8. Adelic toric metrics [02VW]

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5.8. Adelic toric metrics

We now turn to the global case. Let $(\mathbb{K},M_{\mathbb{K}})$ be an adelic field (Definition 2.47). We fix a complete fan $\Sigma$ in $N_{\mathbb{R}}$ and a virtual support function $\Psi$ on $\Sigma$ . Let $(L,s)$ be the associated toric line bundle and section. If $X$ is a variety over $\mathbb{K}$ and $v\in M_{\mathbb{K}}$ we will denote by $X^{{\text{\rm an}},v}$ its analytification with respect to $v$ . Analogously $\mathbb{S}^{{\text{\rm an}},v}$ will denote the compact subtorus of $\mathbb{T}^{{\text{\rm an}},v}$ .

Definition 5.84.

A toric metric on $L$ is a family $(\|\cdot\|_{v})_{v\in M_{\mathbb{K}}}$ , where $\|\cdot\|_{v}$ is a toric metrics on $L_{v}^{{\text{\rm an}}}$ . A toric metric is called adelic if $\psi_{\|\cdot\|_{v}}=\Psi$ for all but finitely many $v$ .

Theorem 5.85.

Let $(\mathbb{K},M_{\mathbb{K}})$ be a global field. A toric metric on $L$ is quasi-algebraic (Definition 2.52) if and only if it is an adelic toric metric.

Proof.

Let $(\|\cdot\|_{v})_{v\in M_{\mathbb{K}}}$ be a metric on $L$ and write ${\overline{L}}=(L,(\|\cdot\|_{v})_{v\in M_{\mathbb{K}}})$ . Suppose first that ${\overline{L}}$ is toric and quasi-algebraic. Let $S\subset M_{\mathbb{K}}$ be a finite set containing the Archimedean places, $\mathbb{K}^{\circ}_{S}$ as in Definition 2.51, $e\geq 1$ an integer and $({\mathcal{X}},{\mathcal{L}})$ a proper model over $\mathbb{K}^{\circ}_{S}$ of $(X_{\Sigma},L^{\otimes e})$ so that $\|\cdot\|_{v}$ is induced by the localization ${\mathcal{L}}_{v}$ for all $v\notin S$ . Over $\mathbb{K}$ , there is an isomorphism from $({\mathcal{X}},{\mathcal{L}})$ to the canonical model $({\mathcal{X}}_{\Sigma},{\mathcal{L}}_{e\Psi})$ . Since $\mathbb{K}^{\circ}_{S}$ is Noetherian, this isomorphism and its inverse are defined over $\mathbb{K}^{\circ}_{S^{\prime}}$ for certain finite subset $S^{\prime}$ containing $S$ . Thus, enlarging the finite set $S$ if necessary, we can suppose without loss of generality that $({\mathcal{X}},{\mathcal{L}})$ agrees with the canonical model $({\mathcal{X}}_{\Sigma},{\mathcal{L}}_{e\Psi})$ . Hence, $\|\cdot\|_{v}=\|\cdot\|_{v,e\Psi}^{1/e}=\|\cdot\|_{v,\Psi}$ for all places $v\notin S$ . In consequence, it is an adelic toric metric.

Conversely, suppose that ${\overline{L}}$ is a toric adelic metrized line bundle. Let $S$ be the union of the set of Archimedean places and $\{v\in M_{\mathbb{K}}|\psi_{v}\neq\Psi\}$ . By definition, this is a finite set. Let $({\mathcal{X}}_{\Sigma},{\mathcal{L}}_{\Psi})$ be the canonical model over $\mathbb{K}^{\circ}_{S}$ of $(X_{\Sigma},L)$ . Then $\|\cdot\|_{v}$ is the metric induced by this model, for all $v\notin S$ . Hence ${\overline{L}}$ is quasi-algebraic. ∎

Corollary 5.86.

Let $L$ be as before.

(1)

There is a bijection between the set of approachable adelic toric metric on $L$ and the set of families of continuous concave functions $\{\psi_{v}\}_{v}$ on $N_{\mathbb{R}}$ such that $|\psi_{v}-\Psi|$ is bounded and $\psi_{v}=\Psi$ for all but finitely many $v$ .
(2)

There is a bijection between the set of approachable adelic toric metric on $L$ and the set of families of continuous concave functions $\{\psi^{\vee}_{v}\}_{v}$ on $\Delta_{\Psi}$ such that $\psi^{\vee}_{v}=0$ for all but finitely many $v$ .

Proof.

This follows from Theorem 5.85 and Theorem 5.73. ∎