5. Metrics and measures on toric varieties [02ST]

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5. Metrics and measures on toric varieties

The aim of this section is to characterize the metrics on a toric line bundle over a toric variety that are, at the same time, invariant under the action of the compact torus and approachable or integrable. Moreover we study the associated measures.

5.1. The variety with corners $X_{\Sigma}(\mathbb{R}_{\geq 0})$

Let $K$ be either $\mathbb{R}$ , $\mathbb{C}$ or a complete field with respect to an absolute value associated to a nontrivial discrete valuation. When $K=\mathbb{R}$ we will use the technique of Remark 2.5 and in the non-Archimedean case we will use the notations of §2.3. Let $\mathbb{T}$ be an $n$ -dimensional split torus over $K$ and let $N$ and $M=N^{\vee}$ be the corresponding lattices. Let $\Sigma$ be a fan in $N_{\mathbb{R}}$ . For each cone $\sigma\in\Sigma$ , we will denote by $X_{\sigma}^{{\text{\rm an}}}$ the complex analytic space $X_{\sigma}(\mathbb{C})$ in Archimedean case or the Berkovich analytic space associated to the scheme $X_{\sigma,K}$ in the non-Archimedean case. These analytic spaces glue together in an analytic space $X_{\Sigma}^{{\text{\rm an}}}$ .

Given any cone $\sigma\in\Sigma$ , we write

X_{\sigma}(\mathbb{R}_{\geq 0})=\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},(\mathbb{R}_{\geq 0},\times)).

On $X_{\sigma}(\mathbb{R}_{\geq 0})$ , we put the coarsest topology such that, for each $m\in M_{\sigma}$ , the map $X_{\sigma}(\mathbb{R}_{\geq 0})\to\mathbb{R}_{\geq 0}$ given by $\gamma\mapsto\gamma(m)$ is continuous. Observe that if $\tau$ is a face of $\sigma$ , then there is a dense open immersion $X_{\tau}(\mathbb{R}_{\geq 0})\hookrightarrow X_{\sigma}(\mathbb{R}_{\geq 0})$ . Hence the topological spaces $X_{\sigma}(\mathbb{R}_{\geq 0})$ glue together to define a topological space $X_{\Sigma}(\mathbb{R}_{\geq 0})$ . This is the variety with corners associated to $X_{\Sigma}$ . Analogously to the algebraic case, one can prove that this topological space is Hausdorff and that the spaces $X_{\sigma}(\mathbb{R}_{\geq 0})$ can be identified with open subspaces of $X_{\Sigma}(\mathbb{R}_{\geq 0})$ satisfying

X_{\sigma}(\mathbb{R}_{\geq 0})\cap X_{\sigma^{\prime}}(\mathbb{R}_{\geq 0})=X_{\sigma\cap\sigma^{\prime}}(\mathbb{R}_{\geq 0}).

For each $\sigma\in\Sigma$ there is a continuous map $\rho_{\sigma}\colon X^{{\text{\rm an}}}_{\sigma}\to X_{\sigma}(\mathbb{R}_{\geq 0})$ . This map is given, in the Archimedean case, by

X^{{\text{\rm an}}}_{\sigma}=\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},(\mathbb{C},\times))\overset{|\cdot|}{\longrightarrow}\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},(\mathbb{R}_{\geq 0},\times))=X^{{\text{\rm an}}}_{\sigma}(\mathbb{R}_{\geq 0}).

While, in the non-Archimedean case, since a point $p\in X_{\sigma}^{{\text{\rm an}}}$ corresponds to a multiplicative seminorm on $K[M_{\sigma}]$ and a point in $X_{\sigma}(\mathbb{R}_{\geq 0})$ corresponds to a semigroup homomorphism from $M_{\sigma}$ to $(\mathbb{R}_{\geq 0},\times)$ , we can define $\rho_{\sigma}(p)$ as the semigroup homomorphism that, to an element $m\in M_{\sigma}$ , corresponds $|\chi^{m}(p)|$ . These maps glue together to define a continuous map $\rho_{\Sigma}:X_{\Sigma}^{{\text{\rm an}}}\to X_{\Sigma}(\mathbb{R}_{\geq 0})$ .

Lemma 5.1.

The map $\rho_{\Sigma}$ satisfies $\rho_{\Sigma}^{-1}(X_{\sigma}(\mathbb{R}_{\geq 0}))=X_{\sigma}^{{\text{\rm an}}}$ .

Proof.

By definition $X_{\sigma}^{{\text{\rm an}}}\subset\rho_{\Sigma}^{-1}(X_{\sigma}(\mathbb{R}_{\geq 0}))$ . For the reverse inclusion we will write only the non-Archimedean case. Assume that $p\in\rho_{\Sigma}^{-1}(X_{\sigma}(\mathbb{R}_{\geq 0}))$ . There is a $\sigma^{\prime}$ with $p\in X_{\sigma^{\prime}}^{{\text{\rm an}}}$ . Let $\tau=\sigma\cap\sigma^{\prime}$ be the common face. Then $p$ is a multiplicative seminorm of $K[M_{\sigma^{\prime}}]$ and we show next that it can be extended to a multiplicative seminorm of $K[M_{\tau}]$ . By [Ful93, §1.2 Proposition 2] there is an element $u\in M_{\sigma^{\prime}}$ such that $M_{\tau}=M_{\sigma^{\prime}}+\mathbb{Z}_{\geq 0}(-u)$ . Hence $K[M_{\tau}]=K[M_{\sigma^{\prime}}+\mathbb{Z}_{\geq 0}(-u)]$ . Since $\rho_{\Sigma}(p)\in X_{\tau}(\mathbb{R}_{\geq 0})$ we have that $|\chi^{u}(p)|\not=0$ . Therefore $p$ extends to a multiplicative seminorm of $K[M_{\tau}]$ . Hence $p\in X^{{\text{\rm an}}}_{\tau}\subset X^{{\text{\rm an}}}_{\sigma}$ . ∎

When $\Sigma$ is complete, the analytic space $X_{\Sigma}^{{\text{\rm an}}}$ is compact, and the map $\rho_{\Sigma}$ is proper. By Lemma 5.1, for each cone $\sigma\in\Sigma$ , the map $\rho_{\sigma}$ is proper. Since every rational cone belongs to a complete fan, the map $\rho_{\Sigma}$ is proper even if $\Sigma$ is not complete. Of particular interest is the case when $\sigma=\{0\}$ . Then $\mathbb{T}^{{\text{\rm an}}}:=X_{0}^{{\text{\rm an}}}$ is an Abelian analytic group, that is, an Abelian group object in the category of analytic spaces. In particular, for any field extension $K^{\prime}$ of $K$ , the set $X_{0}^{{\text{\rm an}}}(K^{\prime})$ is an Abelian group. Also $\mathbb{T}(\mathbb{R}_{\geq 0}):=X_{0}(\mathbb{R}_{\geq 0})\simeq(\mathbb{R}_{\geq 0})^{n}$ is a topological Abelian group. Moreover, $\mathbb{T}^{{\text{\rm an}}}$ acts on $X^{{\text{\rm an}}}_{\Sigma}$ , $\mathbb{T}(\mathbb{R}_{\geq 0})$ acts on $X_{\Sigma}(\mathbb{R}_{\geq 0})$ and the map $\rho_{\Sigma}$ is equivariant with respect to these actions. The kernel of the map $\rho_{0}$ is a closed subgroup, that we call the compact torus of $\mathbb{T}^{{\text{\rm an}}}$ and we denote by $\mathbb{S}^{{\text{\rm an}}}$ . In the Archimedean case it is isomorphic to $(S^{1})^{n}$ , while in the non-Archimedean case it is the compact torus of Example 2.8. In fact, the fibres of the map $\rho_{\Sigma}$ are orbits under the action of $\mathbb{S}^{{\text{\rm an}}}$ . Therefore the space $X_{\sigma}(\mathbb{R}_{\geq 0})$ is the quotient of $X_{\sigma}^{{\text{\rm an}}}$ by the action of the closed subgroup $\mathbb{S}^{{\text{\rm an}}}$ . We warn the reader that the compact topological space underlying $\mathbb{S}^{{\text{\rm an}}}$ is not an abstract group (see [Ber90, Chapter 5]).

The maps $\rho_{\sigma}$ , $\sigma\in\Sigma$ , have canonical sections that we denote $\theta_{\sigma}$ . These sections glue together to give a section $\theta_{\Sigma}$ of $\rho_{\Sigma}$ . In the Archimedean case $\theta_{\sigma}$ is induced by the semigroup inclusion $\mathbb{R}_{\geq 0}\subset\mathbb{C}$ . In the non-Archimedean case $\theta_{\sigma}$ is defined by the following result.

Proposition-Definition 5.2.

Assume that we are in the non-Archimedean case. For each $\gamma\in\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},\mathbb{R}_{\geq 0})$ , the seminorm that, to a function $\sum\alpha_{m}\chi^{m}\in K[M_{\sigma}]$ assigns the value $\sup_{m}(|\alpha_{m}|\gamma(m))$ , is a multiplicative seminorm on $K[M_{\sigma}]$ that extends the norm of $K$ . Therefore it determines a point of $X^{{\text{\rm an}}}_{\sigma}$ that we denote as $\theta_{\sigma}(\gamma)$ . The maps $\theta_{\sigma}$ are injective, continuous and proper. Moreover, they glue together to define a map

\theta_{\Sigma}\colon X_{\Sigma}(\mathbb{R}_{\geq 0})\longrightarrow X_{\Sigma}^{{\text{\rm an}}}

that is injective, continuous and proper. Every point in the image of $\theta_{\Sigma}$ is fixed under the action of $\mathbb{S}^{{\text{\rm an}}}$ .

Proof.

The fact that the seminorm $\theta_{\sigma}(\gamma)$ extends the norm of $K$ is clear. Let now $f=\sum_{m}\alpha_{m}\chi^{m}$ and $g=\sum_{l}\beta_{l}\chi^{l}$ and write $fg=\sum_{k}\varepsilon_{k}\chi^{k}$ with $\varepsilon_{k}=\sum_{m+l=k}\alpha_{m}\beta_{l}$ . Then, since the absolute value of $K$ is ultrametric,

\sup_{k\in M_{\sigma}}(|\varepsilon_{k}|\gamma(k))\leq\sup_{m\in M_{\sigma}}(|\alpha_{m}|\gamma(m))\sup_{l\in M_{\sigma}}(|\beta_{l}|\gamma(l)).

Let $M_{f}=\{m\in M_{\sigma}|\sup_{m^{\prime}}(|\alpha_{m^{\prime}}|\gamma(m^{\prime}))=|\alpha_{m}|\gamma(m)\}$ . We define $M_{g}$ analogously. Let $r$ be a vertex of the Minkowski sum $\operatorname{conv}(M_{f})+\operatorname{conv}(M_{g})$ . Then there is a unique decomposition $r=m_{r}+l_{r}$ with $m_{r}\in M_{f}$ and $l_{r}\in M_{g}$ . Hence $\varepsilon_{r}=\alpha_{m_{r}}\beta_{l_{r}}$ . Thus

\sup_{k\in M_{\sigma}}(|\varepsilon_{k}|\gamma(k))\geq|\varepsilon_{r}|\gamma(r)=\sup_{m\in M_{\sigma}}(|\alpha_{m}|\gamma(m))\sup_{l\in M_{\sigma}}(|\beta_{l}|\gamma(l)).

Thus $\theta_{\sigma}(\gamma)(fg)=\theta_{\sigma}(\gamma)(f)\theta_{\sigma}(\gamma)(g)$ . Hence, it is a multiplicative.

We show next that the map $\theta_{\sigma}$ is continuous. The topology of $X^{{\text{\rm an}}}_{\sigma}$ is the coarsest topology that makes the functions $p\to|f(p)|$ continuous for all $f\in K[M_{\sigma}]$ . Thus to show that $\theta_{\sigma}$ is continuous it is enough to show that the map $\gamma\to|f(\theta_{\sigma}(\gamma))|$ is continuous on $X_{\sigma}(\mathbb{R}_{\geq 0})=\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},\mathbb{R}_{\geq 0})$ . The topology of $X_{\sigma}(\mathbb{R}_{\geq 0})$ is the coarsest topology such that, for each $m\in M_{\sigma}$ , the map $\gamma\to\gamma(m)$ is continuous. Since, for $f=\sum_{m\in M_{\sigma}}\alpha_{m}\chi^{m}$ , we have that

|f(\theta_{\sigma}(\gamma))|=\max(|\alpha_{m}|\gamma(m)),

we obtain that $\theta_{\sigma}$ is continuous. Since each $\theta_{\sigma}$ is a section of $\rho_{\sigma}$ , they are injective.

The fact that the maps $\theta_{\sigma}$ glue together to give a continuous map $\theta_{\Sigma}$ and that $\theta_{\Sigma}$ is a section of $\rho_{\Sigma}$ follows easily from the definitions. This implies in particular that $\theta_{\Sigma}$ is injective. When $\Sigma$ is complete, since $X_{\Sigma}(\mathbb{R}_{\geq 0})$ is compact and $X^{{\text{\rm an}}}_{\Sigma}$ is Hausdorff, the map $\theta_{\Sigma}$ is proper. We deduce that the map $\theta_{\Sigma}$ is proper in general, by using the same argument that shows that the function $\rho_{\Sigma}$ is proper.

The last assertion is clear from the definition of $\theta_{\sigma}(\gamma)$ . ∎

Let now

(5.3)

\lambda_{K}=\begin{cases}1,&\text{ if }K=\mathbb{C},\\ -\log|\varpi|,&\text{ otherwise.}\end{cases}

and denote by ${\operatorname{\mathbf{e}}}_{K}\colon\mathbb{R}\to\mathbb{R}_{>0}$ the map $u\mapsto\exp(-\lambda_{K}u)$ . This map induces an homeomorphism $N_{\mathbb{R}}\rightarrow X_{0}(\mathbb{R}_{>0})$ that we also denote by ${\operatorname{\mathbf{e}}}_{K}$ .

In the non-Archimedean case, the map ${\operatorname{val}}\colon\mathbb{T}(K)\to N$ of Definition 4.71, can be extended to a map $\mathbb{T}^{{\text{\rm an}}}\to N_{\mathbb{R}}$ that we denote ${\operatorname{val}}_{K}$ or, when $K$ is clear from the context by ${\operatorname{val}}$ . For each $p\in X^{{\text{\rm an}}}_{0}$ we denote by ${\operatorname{val}}_{K}(p)\in\operatorname{Hom}_{\operatorname{sg}}(M,\mathbb{R})=N_{\mathbb{R}}$ the morphism

(5.4)

m\longmapsto\langle m,{\operatorname{val}}_{K}(p)\rangle=\frac{-\log|\chi^{m}(p)|}{\lambda_{K}}.

In the Archimedean case we will denote by ${\operatorname{val}}_{\mathbb{C}}$ or simply by ${\operatorname{val}}$ the map defined by the same equation. Then, the diagram

(5.5)

is commutative.

The map ${\operatorname{\mathbf{e}}}_{K}$ allows us to see $X_{\Sigma}(\mathbb{R}_{\geq 0})$ as a partial compactification on $N_{\mathbb{R}}$ . Following [AMRT75, Chapter I, §1] we can give another description of the topology of $X_{\Sigma}(\mathbb{R}_{\geq 0})$ . For $\sigma\in\Sigma$ , we denote

N_{\sigma}=\coprod_{\tau\text{ face of }\sigma}N(\tau)_{\mathbb{R}}.

We choose a positive definite bilinear pairing in $N_{\mathbb{R}}$ . Hence we can identify the quotient spaces $N(\tau)_{\mathbb{R}}$ with subspaces of $N_{\mathbb{R}}$ , that, for simplicity, we will denote also by $N(\tau)_{\mathbb{R}}$ . For a point $u\in N(\tau)_{\mathbb{R}}$ , let $U\subset N(\tau)_{\mathbb{R}}$ be a neighbourhood of $u$ . For each $\tau^{\prime}$ face of $\tau$ , $\tau$ induces a cone $\pi_{\tau^{\prime}}(\tau)$ contained in $N(\tau^{\prime})_{\mathbb{R}}$ . If $p\in\tau$ its image $\pi_{\tau^{\prime}}(p)$ in $N(\tau^{\prime})_{\mathbb{R}}$ , is contained in $\pi_{\tau^{\prime}}(\tau)$ . We write

(5.6)

W(\tau,U,p)=\coprod_{\tau^{\prime}\text{ face of }\tau}\pi_{\tau^{\prime}}(U+p+\tau).

Moving $U$ and $p$ we obtain a basis of neighbourhoods of $u$ in $N_{\sigma}$ . This defines a topology on $N_{\sigma}$ such that the map ${\operatorname{\mathbf{e}}}_{K}\colon N_{\mathbb{R}}\to X_{\sigma}(\mathbb{R}_{\geq 0})$ extends to a homeomorphism $N_{\sigma}\to X_{\sigma}(\mathbb{R}_{\geq 0})$ .

We write

N_{\Sigma}=\coprod_{\sigma\in\Sigma}N(\sigma)_{\mathbb{R}},

and put in $N_{\Sigma}$ the topology that makes $\{N_{\sigma}\}_{\sigma\in\Sigma}$ an open cover. Then the map ${\operatorname{\mathbf{e}}}_{K}$ extends to a homeomorphism between $N_{\Sigma}$ and $X_{\Sigma}(\mathbb{R}_{\geq 0})$ and the map ${\operatorname{val}}_{K}$ extends to a proper continuous map $X^{{\text{\rm an}}}_{\Sigma}\to N_{\Sigma}$ such that the diagram

(5.7)

is commutative.

Remark 5.8.

In case we are given a strictly concave support function $\Psi$ on a fan $\Sigma$ , then $N_{\Sigma}$ is homeomorphic to the polytope $\Delta_{\Psi}$ introduced in §4.4. An homeomorphism is obtained as the composition of ${\operatorname{\mathbf{e}}}_{K}$ with the moment map $\mu\colon X_{\Sigma}(\mathbb{R}_{\geq 0})\rightarrow\Delta_{\Psi}$ induced by $\Psi$ :

\begin{matrix}N_{\Sigma}&\mathrel{\mathop{\kern 0.0pt\longrightarrow}\limits^{{\operatorname{\mathbf{e}}}_{K}}}&X_{\Sigma}(\mathbb{R}_{\geq 0})&\mathrel{\mathop{\kern 0.0pt\longrightarrow}\limits^{\mu}}&\Delta_{\Psi}\\[5.69054pt] u&\longmapsto&{\operatorname{\mathbf{e}}}_{K}(u)&\longmapsto&\frac{\sum\exp(-\lambda_{K}\langle m,u\rangle)m}{\sum\exp(-\lambda_{K}\langle m,u\rangle)}\end{matrix}

where the sums in the last expression are over the elements $m\in M\cap\Delta_{\Psi}$ .

We end this section stating the functorial properties of the space $X_{\Sigma}(\mathbb{R}_{\geq 0})$ . The proofs are left to the reader. Let $N$ and $\Sigma$ be as before and $\sigma\in\Sigma$ . Recall that the associated closed subvariety $V(\sigma)$ is canonically isomorphic to the toric variety $X_{\Sigma(\sigma)}$ .

Proposition 5.9.

The natural map $N(\sigma)_{\mathbb{R}}\hookrightarrow N_{\sigma}$ extends to a continuous map $X_{\Sigma(\sigma)}(\mathbb{R}_{\geq 0})\to X_{\Sigma}(\mathbb{R}_{\geq 0})$ . Moreover, there are commutative diagrams

Let $N_{1}$ and $N_{2}$ be lattices and let $\Sigma_{1}$ and $\Sigma_{2}$ be complete fans in $N_{1,\mathbb{R}}$ and $N_{2,\mathbb{R}}$ respectively. Let $H\colon N_{1}\to N_{2}$ be a linear map such that, for each cone $\sigma_{1}\in\Sigma_{1}$ , there is a cone $\sigma_{2}\in\Sigma_{2}$ with $H(\sigma_{1})\subset\sigma_{2}$ . Let $p\in X_{\Sigma_{2},0}(K)$ and let $A\colon N_{1,\mathbb{R}}\to N_{2,\mathbb{R}}$ be the affine map $A=H+{\operatorname{val}}(p)$ .

Proposition 5.10.

The affine map $A\colon N_{1,\mathbb{R}}\to N_{2,\mathbb{R}}$ extends to a continuous map $X_{\Sigma_{1}}(\mathbb{R}_{\geq 0})\to X_{\Sigma_{2}}(\mathbb{R}_{\geq 0})$ that we also denote by $\varphi_{p,H}$ . Moreover, there are commutative diagrams

5.2. Toric metrics

From now on we assume that $\Sigma$ is complete. Let $L$ be a toric line bundle on $X_{\Sigma}$ and let $s$ be a toric section of $L$ (Definition 4.19). By Theorem 4.22 and Theorem 4.18, we can find a virtual support function $\Psi$ on $\Sigma$ such that there is an isomorphism $L\simeq\mathcal{O}(D_{\Psi})$ that sends $s$ to $s_{\Psi}$ . The algebraic line bundle $L$ defines an analytic line bundle $L^{{\text{\rm an}}}$ on $X_{\Sigma}^{{\text{\rm an}}}$ . Let ${\overline{L}}=(L,\|\cdot\|)$ , where $\|\cdot\|$ is a metric on $L^{{\text{\rm an}}}$ .

Every toric object has a certain invariance property with respect to the action of $\mathbb{T}$ . This is also the case for metrics. Since $\mathbb{T}^{{\text{\rm an}}}$ is non compact, we can not ask for a metric to be $\mathbb{T}^{{\text{\rm an}}}$ -invariant, but we can impose $\mathbb{S}^{{\text{\rm an}}}$ -invariance. We need a preliminary result.

Proposition 5.11.

Let $L$ be a toric line bundle on $X_{\Sigma}$ and let $\|\cdot\|$ be a metric on $L^{{\text{\rm an}}}$ . If there is a toric section $s_{0}$ such that the function $p\mapsto\|s_{0}(p)\|$ is $\,\mathbb{S}^{{\text{\rm an}}}$ -invariant, then, for every toric section $s$ , the function $p\mapsto\|s(p)\|$ is $\,\mathbb{S}^{{\text{\rm an}}}$ -invariant.

Proof.

If $s$ and $s^{\prime}$ are two toric sections, then there is an element $m\in M$ such that $s^{\prime}=\chi^{m}s$ . Since for any element $t\in\mathbb{S}^{{\text{\rm an}}}$ we have $|\chi^{m}(t)|=1$ , if the function $\|s(p)\|$ is $\mathbb{S}^{{\text{\rm an}}}$ -invariant, then the function $\|s^{\prime}(p)\|=\|\chi^{m}(p)s(p)\|$ is also $\mathbb{S}^{{\text{\rm an}}}$ -invariant. ∎

Definition 5.12.

Let $L$ be a toric line bundle on $X_{\Sigma}$ . A metric on $L^{{\text{\rm an}}}$ is called toric if, for any toric section $s$ of $L$ over $X_{0}$ , the function $p\longmapsto\|s(p)\|$ is $\mathbb{S}^{{\text{\rm an}}}$ -invariant.

To the metrized line bundle ${\overline{L}}$ and the section $s$ we associate the function $g_{{\overline{L}},s}\colon X_{0}^{{\text{\rm an}}}\to\mathbb{R}$ given by $g_{{\overline{L}},s}(p)=\log(\|s(p)\|)/\lambda_{K}$ . In the Archimedean case, the function $g_{{\overline{L}},s}$ is $-1/2$ times the usual Green function associated to the metrized line bundle ${\overline{L}}$ and the section $s$ . The metric $\|\cdot\|$ is toric if and only if the function $g_{{\overline{L}},s}$ is $\mathbb{S}^{{\text{\rm an}}}$ -invariant. In this case we can form the commutative diagram

(5.13)

The dashed arrow exists as a continuous function because $\rho_{0}$ , hence ${\operatorname{val}}_{K}$ , is a proper surjective map and, by $\mathbb{S}^{{\text{\rm an}}}$ -invariance, $g_{{\overline{L}},s}$ is constant along the fibres. This justifies the following definition.

Definition 5.14.

Let $L$ be a toric line bundle, $s$ a toric section of $L$ and let $\|\cdot\|$ be a toric metric. Denote ${\overline{L}}=(L,\|\cdot\|)$ . We define the function $\psi_{{\overline{L}},s}\colon N_{\mathbb{R}}\to\mathbb{R}$ by

(5.15)

\psi_{{\overline{L}},s}(u)=\frac{\log\|s(p)\|}{\lambda_{K}}

for any $p\in X_{0}^{{\text{\rm an}}}$ with ${\operatorname{val}}_{K}(p)=u$ . When the line bundle and the section are clear from the context, we will alternatively denote this function as $\psi_{\|\cdot\|}$ .

Proposition 5.16.

Let $\Psi$ be a virtual support function on $\Sigma$ , $L=\mathcal{O}(D_{\Psi})$ and $s=s_{\Psi}$ . Then the correspondence $\|\cdot\|\mapsto\psi_{\|\cdot\|}$ determines a bijection between the set of toric metrics on $L^{{\text{\rm an}}}$ and the set of continuous functions $\psi$ on $N_{\mathbb{R}}$ with the property that $\psi-\Psi$ can be extended to a continuous function on $N_{\Sigma}$ . The metric associated to a function $\psi$ will be denoted $\|\cdot\|_{\psi}$ .

Proof.

Let $\|\cdot\|$ be a toric metric on $L^{{\text{\rm an}}}$ . Since $s$ is a regular nowhere vanishing section on $X_{0}^{{\text{\rm an}}}$ , $\psi_{\|\cdot\|}$ is a well defined continuous function on $N_{\mathbb{R}}$ . Let $\{m_{\sigma}\}$ be a set of defining vectors of $\Psi$ . For each cone $\sigma\in\Sigma$ , the section $\chi^{m_{\sigma}}s$ is a regular nowhere vanishing section on $X_{\sigma}^{{\text{\rm an}}}$ . Therefore $\log(\|(\chi^{m_{\sigma}}s)(p)\|)$ is a continuous function on $X_{\sigma}^{{\text{\rm an}}}$ that is $\mathbb{S}^{{\text{\rm an}}}$ -invariant. So it defines a continuous function on $X_{\sigma}(\mathbb{R}_{\geq 0})$ . By equation (5.4),

	$\displaystyle\psi_{\\|\cdot\\|}({\operatorname{val}}(p))-m_{\sigma}({\operatorname{val}}(p))$	$\displaystyle=\frac{1}{\lambda_{K}}\left(\log(\\|s(p)\\|)-\log(\|\chi^{-m_{\sigma}}(p)\|)\right)$
		$\displaystyle=\frac{1}{\lambda_{K}}\log(\\|(\chi^{m_{\sigma}}s)(p)\\|).$

Therefore $\psi_{\|\cdot\|}-m_{\sigma}$ extends to a continuous function on $N_{\sigma}\simeq X_{\sigma}(\mathbb{R}_{\geq 0})$ . If we see that $\Psi-m_{\sigma}$ extends also to a continuous function on $N_{\sigma}$ we will be able to extend $\psi_{\|\cdot\|}-\Psi$ to a continuous function on $N_{\sigma}$ for every $\sigma\in\Sigma$ and therefore to $N_{\Sigma}$ .

Let $\tau$ be a face of $\sigma$ and let $u\in N(\tau)_{\mathbb{R}}$ . Let $W(\tau,U,p)$ be a neighbourhood of $u$ as in (5.6). By taking $U$ small enough and $p$ big enough we can assume that $W(\tau,U,p)\cap N_{\mathbb{R}}$ is contained in the set of cones that have $\tau$ as a face. Since $\Psi$ and $m_{\sigma}$ agree when restricted to $\sigma$ (hence when restricted to $\tau$ ) it follows that, if $w+t\in W(\tau,U,p)\cap N_{\mathbb{R}}$ with $w\in U$ and $t\in p+\tau$ , then $(\Psi-m_{\sigma})(w+t)$ only depends on $w$ and not on $t$ . Hence it can be extended to a continuous function on the whole $W(\tau,U,p)$ . By moving $\tau$ , $u$ , $U$ and $p$ we see that it can be extended to a continuous function on $N_{\sigma}$ .

Let now $\psi$ be a function on $N_{\mathbb{R}}$ such that $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$ . We define a toric metric $\|\cdot\|_{\psi}$ on $L^{{\text{\rm an}}}$ over the set $X_{0}^{{\text{\rm an}}}$ by the formula

\|s(p)\|_{\psi}=\exp(\lambda_{K}\psi({\operatorname{val}}_{K}(p))).

Then, by the argument before, $\psi-m_{\sigma}$ extends to a continuous function on $N_{\sigma}$ , which proves that $\|\cdot\|_{\psi}$ extends to a metric over $X_{\sigma}^{{\text{\rm an}}}$ . Varying $\sigma\in\Sigma$ we obtain that $\|\cdot\|_{\psi}$ extends to a metric over $X_{\Sigma}^{{\text{\rm an}}}$ . ∎

Corollary 5.17.

For any toric metric $\|\cdot\|$ , the function $|\psi_{\|\cdot\|}-\Psi|$ is bounded.

Proof.

Since we are assuming that $\Sigma$ is complete, the space $N_{\Sigma}\simeq X_{\Sigma}(\mathbb{R}_{\geq 0})$ is compact. Thus the corollary follows from Proposition 5.16. ∎

Example 5.18.

With the notation in Example 3.65, consider the standard simplex $\Delta^{n}$ with fan $\Sigma=\Sigma_{\Delta^{n}}$ and support function $\Psi=\Psi_{\Delta^{n}}$ . The corresponding toric variety is $X_{\Sigma}=\mathbb{P}^{n}$ with toric line bundle $L_{\Psi}=\mathcal{O}(1)$ and toric section $s_{\Psi}=s_{\infty}$ .

(1)

The canonical metrics $\|\cdot\|_{{\operatorname{can}}}$ in examples 2.25 and 2.32 are toric and both correspond to the function $\psi_{\|\cdot\|_{{\operatorname{can}}}}=\Psi$ .
(2)

The Fubini-Study metric $\|\cdot\|_{{\operatorname{FS}}}$ in Example 2.2 is also toric and corresponds to the differentiable function $\psi_{\|\cdot\|_{{\operatorname{FS}}}}=f_{{\operatorname{FS}}}$ introduced in Example 3.53.

Proposition 5.19.

The correspondence $({\overline{L}},s)\mapsto\psi_{{\overline{L}},s}$ satisfies the following properties.

(1)

Let ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=1,2$ , be toric line bundles equipped with toric metrics and let $s_{i}$ be a toric section of $L_{i}$ . Then

$\psi_{{\overline{L}}_{1}\otimes{\overline{L}}_{2},s_{1}\otimes s_{2}}=\psi_{{\overline{L}}_{1},s_{1}}+\psi_{{\overline{L}}_{2},s_{2}}.$
(2)

Let ${\overline{L}}=(L,\|\cdot\|)$ be a toric line bundle equipped with a toric metric and let $s$ be a toric section of $L$ . Then

$\psi_{{\overline{L}}^{-1},s^{-1}}=-\psi_{{\overline{L}},s}.$

Proof.

This follows easily from the definitions. ∎

A consequence of Proposition 5.16 is that every toric line bundle has a distinguished metric.

Proposition-Definition 5.20.

Let $\Sigma$ be a complete fan, $X_{\Sigma}$ the corresponding toric variety, and $L$ a toric line bundle on $X_{\Sigma}$ . Let $s$ be a toric section of $L$ and $\Psi$ the virtual support function on $\Sigma$ associated to $(L,s)$ by theorems 4.22 and 4.18. The metric on $L^{{\text{\rm an}}}$ associated to the function $\Psi$ by Proposition 5.16 only depends on the structure of toric line bundle of $L$ . This metric is called the canonical metric of $L^{{\text{\rm an}}}$ and is denoted $\|\cdot\|_{{\operatorname{can}}}$ . We write ${\overline{L}}^{{\operatorname{can}}}=(L,\|\cdot\|_{{\operatorname{can}}})$ .

Proof.

Let $s^{\prime}$ be another toric section of $L$ . Then there is an element $m\in M$ such that $s^{\prime}=\chi^{m}s$ . The corresponding virtual support function is $\Psi^{\prime}=\Psi-m$ . Denote by $\|\cdot\|$ and $\|\cdot\|^{\prime}$ the metrics associated to $s,\Psi$ and to $s^{\prime},\Psi^{\prime}$ respectively. Then

\|s(p)\|^{\prime}=\|\chi^{-m}s^{\prime}(p)\|^{\prime}=\operatorname{e}^{\lambda_{K}(m+\Psi^{\prime})({\operatorname{val}}(p))}=\operatorname{e}^{\lambda_{K}\Psi({\operatorname{val}}(p))}=\|s(p)\|.

Thus both metrics agree. ∎

The canonical metrics $\|\cdot\|_{{\operatorname{can}}}$ in examples 2.25 and 2.32 are particular cases of the canonical metric of Proposition-Definition 5.20.

Proposition 5.21.

The canonical metric is compatible with the tensor product of line bundles.

(1)

Let $L_{i}$ , $i=1,2$ , be toric line bundles. Then ${\overline{L_{1}\otimes L_{2}}}^{{\operatorname{can}}}={\overline{L_{1}}}^{{\operatorname{can}}}\otimes{\overline{L_{2}}}^{{\operatorname{can}}}$ .
(2)

Let $L$ be a toric line bundle. Then ${\overline{L^{-1}}}^{{\operatorname{can}}}=({\overline{L}}^{{\operatorname{can}}})^{-1}$ .

Proof.

This follows easily from the definitions. ∎

Next we describe the behaviour of the correspondence of Proposition 5.16 with respect to equivariant morphisms. We start with the case of orbits. Let $\Sigma$ be a complete fan in $N$ and $\Psi$ a virtual support function on $\Sigma$ . Let $L$ and $s$ be the associated toric line bundle and toric section, and $\{m_{\sigma}\}_{\sigma\in\Sigma}$ a set of defining vectors of $\Psi$ . Let $\sigma\in\Sigma$ and let $V(\sigma)$ be the corresponding closed subvariety. As in Proposition 4.34, the restriction of $L$ to $V(\sigma)$ is a toric line bundle. Since $V(\sigma)$ and $\operatorname{div}(s)$ may not intersect properly we can not restrict $s$ directly to $V(\sigma)$ . By contrast, $D_{\Psi-m_{\sigma}}=\operatorname{div}(\chi^{m_{\sigma}}s)$ intersects properly $V(\sigma)$ and we can restrict the section $\chi^{m_{\sigma}}s$ to $V(\sigma)$ to obtain a toric section of $\mathcal{O}(D_{(\Psi-m_{\sigma})(\sigma)})\simeq L\mid_{V(\sigma)}$ . Denote $\iota\colon V(\sigma)\to X_{\Sigma}$ the closed immersion. For short, we write $s^{\prime}=\chi^{m_{\sigma}}s$ . Then $\iota^{\ast}s^{\prime}$ is a nowhere vanishing section on $O(\sigma)$ . Recall that $V(\sigma)$ has a structure of toric variety given by the fan $\Sigma(\sigma)$ on $N(\sigma)$ (Proposition 4.6). The principal open subset of $V(\sigma)$ is the orbit $O(\sigma)$ .

Let $\|\cdot\|$ be a toric metric on $L^{{\text{\rm an}}}$ and write ${\overline{L}}=(L,\|\cdot\|)$ . By the proof of Proposition 5.16, the function $\psi_{{\overline{L}},s}-m_{\sigma}=\psi_{{\overline{L}},s^{\prime}}$ can be extended to a continuous function on $N_{\sigma}$ that we denote ${\overline{\psi}}_{{\overline{L}},s^{\prime}}$ .

Proposition 5.22.

The function $\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}\colon N(\sigma)_{\mathbb{R}}\to\mathbb{R}$ agrees with the restriction of ${\overline{\psi}}_{{\overline{L}},s^{\prime}}$ to $N(\sigma)_{\mathbb{R}}\subset N_{\sigma}$ .

Proof.

The section $s^{\prime}$ is a nowhere vanishing section over $X_{\Sigma,\sigma}$ . Therefore, the function $g_{{\overline{L}},s^{\prime}}\colon X^{{\text{\rm an}}}_{\Sigma,\sigma}\to\mathbb{R}$ of diagram (5.13) can be extended to a continuous function on $X_{\Sigma,\sigma}$ that we also denote $g_{{\overline{L}},s^{\prime}}$ . By the definition of the inverse image of a metric, there is a commutative diagram

Then the result is a consequence of the definition of $\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}$ and of the commutativity of the diagram

that follows from Proposition 5.9. ∎

Corollary 5.23.

Let ${\overline{L}}$ be a toric line bundle on $X_{\Sigma}$ equipped with the canonical metric, let $\sigma\in\Sigma$ and $\iota\colon V(\sigma)\to X_{\Sigma}$ the closed immersion. Then the restriction $\iota^{\ast}{\overline{L}}$ is a toric line bundle equipped with the canonical metric.

Proof.

Choose a toric section $s$ of $L$ whose divisor meets $V(\sigma)$ properly. Let $\Psi$ be the corresponding virtual support function. The condition of proper intersection is equivalent to $\Psi|_{\sigma}=0$ . Then $\Psi$ extends to a continuous function ${\overline{\Psi}}$ on $N_{\sigma}$ and the restriction of ${\overline{\Psi}}\to N(\sigma)$ is equal to $\Psi(\sigma)$ . Hence the result follows from Proposition 5.22. ∎

We end with the case of an equivariant morphism whose image intersect the principal open subset. Let $N_{i}$ , $\Sigma_{i}$ , $i=1,2$ , $H$ , $p$ and $A$ be as in Proposition 5.10. Let $\Psi_{2}$ be a virtual support function on $\Sigma_{2}$ and let $\Psi_{1}=\Psi_{2}\circ H$ . This is a virtual support function on $\Sigma_{1}$ . Let $(L_{i},s_{i})$ be the corresponding toric line bundles and sections. By Proposition 4.35 and Theorem 4.22, there is an isomorphism $\varphi_{p.H}^{\ast}L_{2}\simeq L_{1}$ that sends $\varphi_{p.H}^{\ast}s_{2}$ to $s_{1}$ . We use this isomorphism to identify them. Let $\|\cdot\|$ be a toric metric on $L_{2}^{{\text{\rm an}}}$ and write ${\overline{L}}_{2}=(L_{2},\|\cdot\|)$ , ${\overline{L}}_{1}=(L_{1},\varphi_{p.H}^{\ast}\|\cdot\|)$ . The following result follows from Proposition 5.10 and is left to the reader.

Proposition 5.24.

The equality $\psi_{{\overline{L}}_{1},s_{1}}=\psi_{{\overline{L}}_{2},s_{2}}\circ A$ holds.

In the case of toric morphism, the canonical metric is stable by inverse image. The following result follows easily from the definitions.

Corollary 5.25.

Assume furthermore that $p=x_{0}$ and so the equivariant morphism $\varphi_{p,H}=\varphi_{H}\colon X_{\Sigma_{1}}\to X_{\Sigma_{2}}$ is a toric morphism. If ${\overline{L}}$ is a toric line bundle on $X_{\Sigma_{2}}$ equipped with the canonical metric, then $\varphi_{H}^{\ast}{\overline{L}}$ is a toric line bundle equipped with the canonical metric.

The inverse image of the canonical metric by an equivariant map does not need to be the canonical metric. In fact, the analogue of Example 4.109 in terms of metrics shows that many different metrics can be obtained as the inverse image of the canonical metric on the projective space.

Example 5.26.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $X_{\Sigma}$ the corresponding toric variety. Recall the description of the projective space $\mathbb{P}^{r}$ as a toric variety given in Example 4.3. Let $H\colon N\to\mathbb{Z}^{r}$ be a linear map such that, for each $\sigma\in\Sigma$ there exist $\tau\in\Sigma_{\Delta^{r}}$ with $H(\sigma)\subset\tau$ . Let $p\in\mathbb{P}^{r}_{0}(K)$ . Then we have an equivariant morphism $\varphi_{p,H}\colon X_{\Sigma}\to\mathbb{P}^{r}$ . Consider the support function $\Psi_{\Delta^{r}}$ on $\Sigma_{\Delta^{r}}$ . Then $L_{\Psi_{\Delta^{r}}}=\mathcal{O}_{\mathbb{P}^{r}}(1)$ . Write $L=\varphi^{\ast}_{p,H}L_{\Psi_{\Delta^{r}}}$ , $s=\varphi^{\ast}_{p,H}s_{\Psi_{\Delta^{r}}}$ and $\Psi=H^{\ast}\Psi_{\Delta^{r}}$ . Thus $(L,s)=(L_{\Psi},s_{\Psi})$ .

Set $A=H+{\operatorname{val}}_{K}(p)$ for the affine map. Let $\|\cdot\|$ be the metric on $L^{{\text{\rm an}}}$ induced by the canonical metric of $\mathcal{O}(D_{\Psi_{\Delta^{r}}})^{{\text{\rm an}}}$ and let $\psi$ be the function associated to it by Proposition 5.16. By Proposition 5.24, $\psi=A^{\ast}\Psi_{\Delta^{r}}$ . This is a piecewise affine concave function on $N_{\mathbb{R}}$ with $\operatorname{rec}(\psi)=\Psi$ that can be made explicit as follows.

Let $\{e_{1},\dots,e_{r}\}$ be the standard basis of $\mathbb{Z}^{r}$ and let $\{e_{1}^{\vee},\dots,e_{r}^{\vee}\}$ be the dual basis. Write $m_{i}=e_{i}^{\vee}\circ H\in M$ and $l_{i}=e_{i}^{\vee}({\operatorname{val}}_{K}(p))\in\mathbb{R}$ . Then

	$\displaystyle\Psi$	$\displaystyle=\min\{0,m_{1},\dots,m_{r}\}$
	$\displaystyle\psi$	$\displaystyle=\min\{0,m_{1}+l_{1},\dots,m_{r}+l_{r}\}$

We want to characterize all the functions that can be obtained with a slight generalization of the previous construction.

Proposition 5.27.

Let $\Sigma$ be a complete fan in $N$ and $\Psi$ a support function on $\Sigma$ . Write $L=L_{\Psi}$ and $s=s_{\Psi}$ . Let $\psi\colon N_{\mathbb{R}}\to\mathbb{R}$ a piecewise affine concave function with $\operatorname{rec}(\psi)=\Psi$ , that has an $H$ -representation

\psi=\min_{i=0,\dots,r}\{m_{i}+l_{i}\},

with $m_{i}\in M_{\mathbb{Q}}$ and $l_{i}\in\mathbb{R}$ in the Archimedean case and $l_{i}\in\mathbb{Q}$ in the non-Archimedean case. Then there is an equivariant morphism $\varphi\colon X_{\Sigma}\to\mathbb{P}^{r}$ , an integer $e>0$ and an isomorphism $L^{\otimes e}\simeq\varphi^{\ast}\mathcal{O}(1)$ such that the metric induced on $L^{{\text{\rm an}}}$ by the canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ agrees with $\|\cdot\|_{\psi}$ .

Proof.

First observe that the condition $l_{i}\in\mathbb{R}$ in the Archimedean case and $l_{i}\in\mathbb{Q}$ in the non-Archimedean case is equivalent to the condition $l_{i}\in\mathbb{Q}\,{\operatorname{val}}_{K}(K^{\times})$ . Let $e>0$ be an integer such that $em_{i}\in M$ and $el_{i}\in{\operatorname{val}}_{K}(K^{\times})$ for $i=0,\dots,r$ .

Consider the linear map $H\colon N_{\mathbb{R}}\to\mathbb{R}^{r}$ given by $H(u)=(em_{i}(u)-em_{0}(u))_{i=1,\dots,r}$ and the affine map $A=H+\boldsymbol{l}$ with $\boldsymbol{l}=(el_{i}-el_{0})_{i=1,\dots,r}$ . By Lemma 3.79,

e\psi=A^{\ast}\Psi_{\Delta^{r}}+em_{0}+el_{0}.

We claim that, for each $\sigma\in\Sigma$ there exists $\sigma_{i_{0}}\in\Sigma_{\Delta^{r}}$ such that $H(\sigma)\subset\sigma_{i_{0}}$ . Indeed, $\Psi(u)=\min_{i}\{m_{i}(u)\}$ . Since $\Psi$ is a support function on $\Sigma$ , for each $\sigma\in\Sigma$ , there exists an $i_{0}$ such that $\Psi(u)=m_{i_{0}}(u)$ for all $u\in\sigma$ . Writing $e_{0}^{\vee}=0$ , this condition implies

\min_{0\leq i\leq r}\{e_{i}^{\vee}(H(u))\}=e_{i_{0}}^{\vee}(H(u))\quad\text{for all }u\in\sigma.

Hence, $H(\sigma)\subset\sigma_{i_{0}}$ , where $\sigma_{i_{0}}\in\Sigma_{\Delta^{r}}$ is the cone $\{v|\min_{0\leq i\leq r}\{e_{i}^{\vee}(v)\}=e_{i_{0}}^{\vee}(v)\}$ and the claim is proved.

Therefore, we can apply Theorem 4.9 and given a point $p\in\mathbb{P}^{r}_{r}(K)$ such that ${\operatorname{val}}_{K}(p)=\boldsymbol{l}$ , there is an equivariant map $\varphi_{p,H}\colon X_{\Sigma}\to\mathbb{P}^{r}$ . By Example 4.44, there is an isomorphism $L^{\otimes e}\simeq\varphi_{p,H}^{*}\mathcal{O}(1)$ and $a\in K^{\times}$ with ${\operatorname{val}}_{K}(a)=l_{0}$ such that $(a^{-1}\chi^{-m_{0}}s)^{\otimes e}$ corresponds to $\varphi_{p,H}^{*}(s_{\Psi_{\Delta^{r}}})$ .

Let ${\overline{L}}$ be the line bundle $L$ equipped with the metric induced by the above isomorphism and the canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ . Then

\psi_{{\overline{L}},s}=\psi_{{\overline{L}},a^{-1}\chi^{-m_{0}}s}+m_{0}+l_{0}=\frac{1}{e}A^{\ast}\Psi_{\Delta^{r}}+m_{0}+l_{0}=\psi,

as stated. ∎

Corollary 5.28.

Let $\psi$ be as in Proposition 5.27. Then the metric $\|\cdot\|_{\psi}$ is approachable.

Proof.

This follows readily from the previous result together with Example 2.32 in the Archimedean case and Example 2.25 in the non-Archimedean case and the fact that the inverse image of an approachable metric is also approachable. ∎

5.3. Smooth metrics and their associated measures

We now discuss the relationship between semipositivity of smooth metrics and concavity of the associated function in the Archimedean case. Moreover we will determine the associated measure.

In this section $K$ is either $\mathbb{R}$ or $\mathbb{C}$ and we fix a lattice $N$ of rank $n$ , a complete fan $\Sigma$ in $N_{\mathbb{R}}$ and a virtual support function $\Psi$ on $\Sigma$ , with $L$ and $s$ the corresponding toric line bundle and section. Let $X_{\Sigma}^{{\text{\rm an}}}$ be the complex analytic space associated to $X_{\Sigma}$ and $L^{{\text{\rm an}}}$ the analytic line bundle associated to $L$ .

Proposition 5.29.

Let $\|\cdot\|$ be a smooth toric metric on $L^{{\text{\rm an}}}$ . Then $\|\cdot\|$ is semipositive if and only if the function $\psi=\psi_{\|\cdot\|}$ is concave.

Proof.

Since the condition of being semipositive is closed, it is enough to check it in the open set $X_{0}^{{\text{\rm an}}}$ . We choose an integral basis of $M=N^{\vee}$ . This determines isomorphisms

X_{0}^{{\text{\rm an}}}\simeq(\mathbb{C}^{\times})^{n},\quad X_{0}(\mathbb{R}_{\geq 0})\simeq(\mathbb{R}_{>0})^{n},\quad N_{\mathbb{C}}\simeq\mathbb{C}^{n},\quad N_{\mathbb{R}}\simeq\mathbb{R}^{n}.

Let $z_{1},\dots,z_{n}$ be the coordinates of $X_{0}^{{\text{\rm an}}}$ and $u_{1},\dots,u_{n}$ the coordinates of $N_{\mathbb{R}}$ determined by these isomorphisms. With these coordinates the map

{\operatorname{val}}\colon X_{0}^{{\text{\rm an}}}\to N_{\mathbb{R}}

is given by

{\operatorname{val}}(z_{1},\dots,z_{n})=\frac{-1}{2}(\log(z_{1}\bar{z}_{1}),\dots,\log(z_{n}\bar{z}_{n})).

As usual, we denote ${\overline{L}}=(L,\|\cdot\|)$ . Set $g=g_{{\overline{L}},s}=\log\|s\|$ . Then, the integral valued first Chern class is given by

(5.30)

\frac{1}{2\pi i}c_{1}(\overline{L})=\frac{1}{\pi i}\partial\bar{\partial}g=\frac{-i}{\pi}\sum_{k,l}\frac{\partial^{2}g}{\partial z_{k}\partial\bar{z}_{l}}\,\text{\rm d}z_{k}\land\,\text{\rm d}\bar{z}_{l}.

The standard orientation of the unit disk $\mathbb{D}\subset\mathbb{C}$ is given by $\,\text{\rm d}x\land\,\text{\rm d}y=(i/2)\,\text{\rm d}z\land\,\text{\rm d}\bar{z}$ . Hence, the metric of ${\overline{L}}$ is semipositive if and only if the matrix $G=(\frac{\partial^{2}g}{\partial z_{k}\partial\bar{z}_{l}})_{k,l}$ is semi-negative definite. Since

(5.31)

\frac{\partial^{2}g}{\partial z_{k}\partial\bar{z}_{l}}=\frac{1}{4z_{k}\bar{z}_{l}}\frac{\partial^{2}\psi}{\partial u_{k}\partial\bar{u}_{l}},

if we write $\operatorname{Hess}(\psi)=(\frac{\partial^{2}\psi}{\partial u_{k}\partial\bar{u}_{l}})_{k,l}$ and $Z=\operatorname{diag}((2z_{1})^{-1},\dots,(2z_{n})^{-1})$ , then $G=\bar{Z}^{t}\operatorname{Hess}(\psi)Z$ . Therefore $G$ is semi-negative definite if and only if $\operatorname{Hess}(\psi)$ is semi-negative definite, hence, if and only if $\psi$ is concave. ∎

The line bundle $L^{{\text{\rm an}}}$ admits a semipositive metric is and only if $\Psi$ is concave. Thus, from now on we assume that $\Psi$ is a support function, that is, a concave support function.

Definition 5.32.

Let $\psi\colon N_{\mathbb{R}}\to\mathbb{R}$ be a concave function such that $|\Psi-\psi|$ is bounded. Let $\mathcal{M}_{M}(\psi)$ be the Monge-Ampère measure associated to $\psi$ and the lattice $M$ . We will denote by ${\overline{\mathcal{M}}}_{M}(\psi)$ the measure on $N_{\Sigma}$ given by

{\overline{\mathcal{M}}}_{M}(\psi)(E)=\mathcal{M}_{M}(\psi)(E\cap N_{\mathbb{R}})

for any Borel subset of $N_{\Sigma}$ .

By its very definition, the measure ${\overline{\mathcal{M}}}_{M}(\psi)$ is bounded with total mass

{\overline{\mathcal{M}}}_{M}(\psi)(N_{\Sigma})=\operatorname{vol}_{M}(\Delta_{\Psi})

and the set $N_{\Sigma}\setminus N_{\mathbb{R}}$ has measure zero.

Theorem 5.33.

Let $\|\cdot\|$ be a semipositive smooth toric metric on $L^{{\text{\rm an}}}$ . Let $c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}$ be the measure defined by ${\overline{L}}$ . Then,

(5.34)

{\operatorname{val}}_{\ast}(c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}})=n!{\overline{\mathcal{M}}}_{M}(\psi),

where ${\operatorname{val}}$ is the map of diagram (5.7). In addition, this measure is uniquely characterized by equation (5.34) and the property of being $\mathbb{S}^{{\text{\rm an}}}$ -invariant.

Proof.

Since the measure $c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}$ is given by a smooth volume form and $X^{{\text{\rm an}}}_{\Sigma}\setminus X^{{\text{\rm an}}}_{0}$ is a set of Lebesgue measure zero, the measure $c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}$ is determined by its restriction to the dense open subset $X_{0}^{{\text{\rm an}}}$ . Thus, to prove equation (5.34) it is enough to show that

(5.35)

{\operatorname{val}}_{\ast}(c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}|_{X^{{\text{\rm an}}}_{0}})=n!\mathcal{M}_{M}(\psi).

We use the coordinate system of the proof of Proposition 5.29. We denote by ${\widetilde{{\operatorname{\mathbf{e}}}}}\colon N_{\mathbb{C}}\to X_{0}(\mathbb{C})$ the map induced by the morphism $\mathbb{C}\to\mathbb{C}^{\times}$ given by $z\mapsto\exp(-z)$ . We write $u_{k}+iv_{k}$ for the complex coordinates of $N_{\mathbb{C}}$ . Then

(5.36)

{\widetilde{{\operatorname{\mathbf{e}}}}}^{\ast}\left(\frac{\,\text{\rm d}z_{k}\land\,\text{\rm d}\bar{z}_{k}}{z_{k}\bar{z}_{k}}\right)=(-2i)\,\text{\rm d}u_{k}\land\,\text{\rm d}v_{k}.

Using now equations (5.30), (5.31) and (5.36), we obtain that,

	$\displaystyle\frac{1}{(2\pi i)^{n}}{\widetilde{{\operatorname{\mathbf{e}}}}}^{\ast}c_{1}({\overline{L}})^{n}$	$\displaystyle={\widetilde{{\operatorname{\mathbf{e}}}}}^{\ast}\left(\frac{1}{(i\pi)^{n}}n!\det G\,\text{\rm d}z_{1}\land\,\text{\rm d}\bar{z}_{1}\land\dots\land\,\text{\rm d}z_{n}\land\,\text{\rm d}\bar{z}_{n}\right)$
		$\displaystyle=\frac{(-1)^{n}}{(2\pi)^{n}}n!\det\operatorname{Hess}(\psi)\,\text{\rm d}u_{1}\land\,\text{\rm d}v_{1}\land\dots\land\,\text{\rm d}u_{n}\land\,\text{\rm d}u_{n}.$

Since the map ${\operatorname{val}}$ is the composition of ${\widetilde{{\operatorname{\mathbf{e}}}}}^{-1}$ with the projection $N_{\mathbb{C}}\to N_{\mathbb{R}}$ , integrating with respect to the variables $v_{1},\dots,v_{n}$ in the domain $[0,2\pi]^{n}$ , taking into account the natural orientation of $\mathbb{C}^{n}$ and the orientation of $N_{\mathbb{R}}$ given by the coordinate system, and the fact that the normalization factor $1/(2\pi i)^{n}$ is implicit in the current $\delta_{X_{\Sigma}}$ , we obtain

{\operatorname{val}}_{\ast}(c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}|_{X_{0}^{{\text{\rm an}}}})=(-1)^{n}n!\det\operatorname{Hess}(\psi)\,\text{\rm d}u_{1}\land\dots\land\,\text{\rm d}u_{n}.

Thus equation (5.35) follows from Proposition 3.94. Finally, the last statement follows from the fact that, in a compact Abelian group there is a unique Haar measure with fixed total volume. ∎

We end this section recalling how to obtain a toric metric from a non-toric one. Let $L$ be a toric line bundle on the toric variety $X_{\Sigma}$ and let $s$ be a toric section. If $\|\cdot\|$ is a smooth, non-necessarily toric, metric, we can average it to obtain a toric metric. This averaging process preserves smoothness and semipositivity. Let $\mu_{\operatorname{Haar}}$ be the Haar measure of $\mathbb{S}^{{\text{\rm an}}}$ of total volume 1. Then we define the metric $\|\cdot\|_{\mathbb{S}}$ over $X_{0}^{{\text{\rm an}}}$ by

(5.37)

\log\|s(p)\|_{\mathbb{S}}=\int_{\mathbb{S}^{{\text{\rm an}}}}\log\|s(t\cdot p)\|\,\text{\rm d}\mu_{\operatorname{Haar}}(t).

Proposition 5.38.

The metric $\|\cdot\|_{\mathbb{S}}$ extends to a toric smooth metric over $X^{{\text{\rm an}}}_{\Sigma}$ . Moreover, if $\|\cdot\|$ is semipositive then $\|\cdot\|_{\mathbb{S}}$ is semipositive.

Proof.

Let $\|\cdot\|^{\prime}$ be any toric smooth metric. Then $\|\cdot\|_{\mathbb{S}}$ extends to a smooth metric if and only if $\log(\|s\|_{\mathbb{S}}/\|s\|^{\prime})$ can be extended to a smooth function on $X_{\Sigma}^{{\text{\rm an}}}$ . But we have

\log(\|s\|_{\mathbb{S}}/\|s\|^{\prime})=\int_{\mathbb{S}^{{\text{\rm an}}}}\log(\|s(t\cdot p)\|/\|s(t\cdot p)\|^{\prime})\,\text{\rm d}\mu_{\operatorname{Haar}}(t)

and the right-hand side can be extended to a smooth function on the whole $X_{\Sigma}^{{\text{\rm an}}}$ . Clearly the metric $\|\cdot\|_{\mathbb{S}}$ is toric. Moreover

c_{1}(L,\|\cdot\|_{\mathbb{S}})=\int_{\mathbb{S}^{{\text{\rm an}}}}t^{\ast}c_{1}(L,\|\cdot\|)\,\text{\rm d}\mu_{\operatorname{Haar}}(t).

Therefore, if $(L,\|\cdot\|)$ is semipositive, then $(L,\|\cdot\|_{\mathbb{S}})$ is semipositive. ∎

5.4. Algebraic metrics from toric models

Next we study some properties of the algebraic metrics that arise from toric models. This kind of metrics will be called toric algebraic metrics. Thus, we assume that $K$ is a complete field with respect to an absolute value associated to a nontrivial discrete valuation. We keep the usual notations. We fix a complete fan $\Sigma$ in $N_{\mathbb{R}}$ .

We begin by studying the relationship between the maps ${\operatorname{val}}$ and ${\operatorname{red}}$ .

Lemma 5.39.

Let $\Pi$ be a complete SCR polyhedral complex of $N_{\mathbb{R}}$ such that $\operatorname{rec}(\Pi)=\Sigma$ . Let ${\mathcal{X}}:={\mathcal{X}}_{\Pi}$ be the model of $X_{\Sigma}$ determined by $\Pi$ . Let $\Lambda\in\Pi$ and $p\in X_{0}^{{\text{\rm an}}}$ . Then ${\operatorname{red}}(p)\in{\mathcal{X}}_{\Lambda}$ if and only if ${\operatorname{val}}_{K}(p)\in\Lambda$ .

Proof.

By the definition of the semigroup ${\widetilde{M}}_{\Lambda}$ , the condition ${\operatorname{val}}(p)\in\Lambda$ holds if and only in $\langle m,{\operatorname{val}}(p)\rangle+l\geq 0$ for all $(m,l)\in{\widetilde{M}}_{\Lambda}$ . This is equivalent to $\log|\chi^{-m}(p)|+\log|\varpi|^{-l}\geq 0$ for all $(m,l)\in{\widetilde{M}}_{\Lambda}$ . In turn, this is equivalent to $|\chi^{m}(p)\varpi^{l}|\leq 1$ , for all $(m,l)\in{\widetilde{M}}_{\Lambda}$ . Hence, ${\operatorname{val}}(p)\in\Lambda$ if and only if $|a(p)|\leq 1$ for all $a\in K^{\circ}[{\mathcal{X}}_{\Lambda}]$ , which is exactly the condition ${\operatorname{red}}(p)\in{\mathcal{X}}_{\Lambda}$ (see (2.12)). ∎

Corollary 5.40.

With the same hypothesis as Lemma 5.39, ${\operatorname{red}}(p)\in O(\Lambda)$ if and only if ${\operatorname{val}}(p)\in\operatorname{ri}(\Lambda)$ .

Proof.

This follows from Lemma 5.39 and the fact that the special fibre is

{\mathcal{X}}_{\Lambda,o}=\coprod_{\Lambda^{\prime}\text{ face of }\Lambda}O(\Lambda^{\prime}),

and $\operatorname{ri}(\Lambda)=\Lambda\setminus\bigcup_{\Lambda^{\prime}\text{ proper face of }\Lambda}\Lambda^{\prime}$ . ∎

Let $\Psi$ be a virtual support function on $\Sigma$ , and $(L,s)$ the corresponding toric line bundle and section. Let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$ such that $\operatorname{rec}(\Pi)=\Sigma$ and let $\psi$ be a rational piecewise affine function on $\Pi$ with $\operatorname{rec}(\psi)=\Psi$ . Let $e>0$ be an integer such that $e\psi$ is an H-lattice function. By Theorem 4.81, the pair $(\Pi,e\psi)$ determines a toric model $({\mathcal{X}}_{\Pi},{\mathcal{L}}_{e\psi},e)$ of $(X_{\Sigma},L)$ . We will write ${\mathcal{L}}={\mathcal{L}}_{e\psi}$ . Definition 2.17 gives us an algebraic metric $\|\cdot\|_{{\mathcal{L}}}$ on $L^{{\text{\rm an}}}$ . In its turn, the metric $\|\cdot\|_{{\mathcal{L}}}$ defines a function $\psi_{\|\cdot\|_{{\mathcal{L}}}}$ . The following proposition closes the circle.

Proposition 5.41.

The equality $\psi_{\|\cdot\|_{{\mathcal{L}}}}=\psi$ holds. Hence $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$ and the metric $\|\cdot\|_{\psi}$ associated to $\psi$ by Proposition 5.16 agrees with $\|\cdot\|_{{\mathcal{L}}}$ .

Proof.

The tensor product $s^{\otimes e}$ defines a rational section of ${\mathcal{L}}$ . Let $\Lambda\in\Pi$ and choose $m_{\Lambda}\in M$ , $l_{\Lambda}\in\mathbb{Z}$ such that $e\psi|_{\Lambda}=m_{\Lambda}+l_{\Lambda}|_{\Lambda}$ . Let $u\in\Lambda$ and $p\in X^{{\text{\rm an}}}_{\Sigma}$ with $u={\operatorname{val}}(p)$ . Then ${\operatorname{red}}(p)\in{\mathcal{X}}_{\Lambda}$ . But in ${\mathcal{X}}_{\Lambda}$ the section $\chi^{m_{\Lambda}}\varpi^{l_{\Lambda}}s^{\otimes e}$ is regular and non-vanishing. Therefore, by Definition 2.17,

\|\chi^{m_{\Lambda}}(p)\varpi^{l_{\Lambda}}s^{\otimes e}(p)\|_{{\mathcal{L}}}=1.

Thus

	$\displaystyle\psi_{\\|\cdot\\|_{{\mathcal{L}}}}(u)$	$\displaystyle=\frac{1}{\lambda_{K}}\log(\\|s(p)\\|_{{\mathcal{L}}})$
		$\displaystyle=\frac{1}{e\lambda_{K}}\log(\|\chi^{-m_{\Lambda}}(p)\varpi^{-l_{\Lambda}}\|)$
		$\displaystyle=\frac{1}{e}(\langle m_{\Lambda},u\rangle+l_{\Lambda})$
		$\displaystyle=\psi(u).$

Therefore $\psi$ agrees with the function associated to the metric $\|\cdot\|_{{\mathcal{L}}}$ . Hence $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$ and the metric $\|\cdot\|_{\psi}$ agrees with $\|\cdot\|_{{\mathcal{L}}}$ . ∎

Example 5.42.

In the non-Archimedean case, the canonical metric of Proposition-Definition 5.20 is the toric algebraic metric induced by the canonical model of Definition 4.76.

Proposition 5.41 imposes a necessary condition for a rational piecewise affine function to determine a model of $(X_{\Sigma},L_{\Psi})$ .

Corollary 5.43.

Let $\Psi$ be a virtual support function on $\Sigma$ and let $\psi$ be a rational piecewise affine function on $N_{\mathbb{R}}$ , with $\operatorname{rec}(\psi)=\Psi$ , such that there exists a complete SCR polyhedral complex $\Pi$ with $\operatorname{rec}(\Pi)=\Sigma$ and $\psi$ piecewise affine on $\Pi$ . Then $\psi-\Psi$ can be extended to a continuous function on $N_{\Sigma}$ .

Proof.

If there exists such a SCR polyhedral complex $\Pi$ , then $\Pi$ and $\psi$ determine a model of $\mathcal{O}(D_{\Psi})$ and hence a toric algebraic metric $\|\cdot\|$ . By Proposition 5.41, $\psi=\psi_{\|\cdot\|}$ and, by the classification of toric metrics in Proposition 5.16, the function $\psi_{\|\cdot\|}-\Psi$ extends to a continuous function on $N_{\Sigma}$ . ∎

Example 5.44.

Let $N=\mathbb{Z}^{2}$ and consider the fan $\Sigma$ generated by $e_{0}=(-1,-1)$ , $e_{1}=(1,0)$ and $e_{2}=(0,1)$ . Then $X_{\Sigma}=\mathbb{P}^{2}$ . The virtual support function $\Psi=0$ corresponds to the trivial line bundle $\mathcal{O}_{\mathbb{P}^{2}}$ . Consider the function

\psi(x,y)=\begin{cases}0,&\text{ if }x\leq 0,\\ x,&\text{ if }0\leq x\leq 1,\\ 1,&\text{ if }1\leq x.\\ \end{cases}

Then $\operatorname{rec}(\psi)=\Psi$ , but $\psi$ does not extend to a continuous function on $N_{\Sigma}$ and therefore it does not determine a model of $(X_{\Sigma},\mathcal{O})$ . By contrast, let $\Sigma^{\prime}$ be the fan obtained subdividing $\Sigma$ by adding the edge corresponding to $e^{\prime}=(0,-1)$ . Then $X_{\Sigma^{\prime}}$ is isomorphic to a blow-up of $\mathbb{P}^{2}$ at one point. The function $\psi$ extends to a continuous function on $N_{\Sigma^{\prime}}$ and it corresponds to a toric model of $(X_{\Sigma^{\prime}},\mathcal{O})$ .

Question 5.45.

Is the condition in Corollary 5.43 also sufficient? In other words, let $N$ , $\Sigma$ and $\Psi$ be as before and let $\psi$ be a rational piecewise affine function on $N_{\mathbb{R}}$ such that $\psi-\Psi$ can be extended to a continuous function on $N_{\Sigma}$ . Does it exists a complete SCR polyhedral complex $\Pi$ with $\operatorname{rec}(\Pi)=\Sigma$ and $\psi$ is piecewise affine on $\Pi$ ?

Remark 5.46.

By the proof of Theorem 4.97 and Corollary 5.43, when $\psi$ is concave, the conditions

(1)

$|\psi-\Psi|$ is bounded;
(2)

$\psi-\Psi$ can be extended to a continuous function on $N_{\Sigma}$ ;
(3)

there exist a complete SCR polyhedral complex $\Pi$ with $\operatorname{rec}(\Pi)=\Sigma$ and $\psi$ piecewise affine on $\Pi$ ;

are equivalent. In particular, the answer to the above question is positive when $\psi$ is concave.

By Theorem 4.97, a rational piecewise affine concave function $\psi$ with $\operatorname{rec}(\psi)=\Psi$ determines an equivalence class of semipositive toric models of $(X_{\Sigma,K},L_{\Psi})$ . As before, every toric model in this class defines an algebraic metric on $L_{\Psi}^{{\text{\rm an}}}$ . Since, by Proposition 2.18, equivalent models give rise to the same metric, this metric only depends on $\psi$ . Then Proposition 5.41 has the following direct consequence.

Corollary 5.47.

Let $\Sigma$ be a complete fan and let $\Psi$ be a support function on $\Sigma$ . Let $\psi$ be a rational piecewise affine concave function on $N_{\mathbb{R}}$ with $\operatorname{rec}(\psi)=\Psi$ and let $\|\cdot\|$ be the metric defined by any model of $(X_{\Sigma},L_{\Psi})$ in the equivalence class determined by $\psi$ . Then the equality $\psi_{\|\cdot\|}=\psi$ holds. So the metric $\|\cdot\|$ agrees with the metric $\|\cdot\|_{\psi}$ of Proposition 5.16. Moreover, the algebraic metric $\|\cdot\|$ is semipositive.

Proof.

The equation $\psi_{\|\cdot\|}=\psi$ is just Proposition 5.41 in the concave case. By the definition of semipositive algebraic metrics and Theorem 4.95 we obtain that $\psi$ concave implies $\|\cdot\|$ semipositive. ∎

We have seen that rational piecewise affine functions give rise to toric algebraic metrics. We now study the converse. Let $g$ be a rational function on $X_{\Sigma}$ . Then we denote by $\psi_{g}\colon N_{\mathbb{R}}\to\mathbb{R}$ the function $\psi_{g}(u)=\frac{1}{\lambda_{K}}\log|g(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))|$ .

Lemma 5.48.

Let $g$ be a rational function on $X_{\Sigma}$ . Then the function $\psi_{g}$ is an H-lattice function (Definition 3.88). In particular it is a piecewise affine function.

Proof.

The function $g$ can be written as $g=\frac{\sum_{m\in M}\alpha_{m}\chi^{m}}{\sum_{m\in M}\beta_{m}\chi^{m}}$ . Then

	$\displaystyle\psi_{g}(u)$	$\displaystyle=\frac{1}{\lambda_{K}}\log\|g(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))\|$
		$\displaystyle=\frac{1}{\lambda_{K}}\log\Bigl\|\sum_{m\in M}\alpha_{m}\chi^{m}(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))\Bigr\|-\frac{1}{\lambda_{K}}\log\Bigl\|\sum_{m\in M}\beta_{m}\chi^{m}(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))\Bigr\|$
		$\displaystyle=\max_{m\in M}\Bigl(\frac{\log\|\alpha_{m}\|}{\lambda_{K}}-\left<m,u\right>\Bigr)-\max_{m\in M}\Bigl(\frac{\log\|\beta_{m}\|}{\lambda_{K}}-\left<m,u\right>\Bigr)$
		$\displaystyle=\max_{m\in M}(-{\operatorname{ord}}(\alpha_{m})-\left<m,u\right>)-\max_{m\in M}(-{\operatorname{ord}}(\beta_{m})-\left<m,u\right>)$
		$\displaystyle=\min_{m\in M}(\left<m,u\right>+{\operatorname{ord}}(\beta_{m}))-\min_{m\in M}(\left<m,u\right>+{\operatorname{ord}}(\alpha_{m})).$

Thus, it is the difference of two H-lattice concave functions. ∎

Theorem 5.49.

Let $\Sigma$ be a complete fan, $\Psi$ a virtual support function on $\Sigma$ and $(L,s)$ the corresponding toric line bundle and section. Let $\|\cdot\|$ be a toric algebraic metric on $L^{{\text{\rm an}}}$ . Then the function $\psi_{\|\cdot\|}$ is rational piecewise affine. If moreover $\psi_{\|\cdot\|}$ is concave, the toric algebraic metric $\|\cdot\|$ is semipositive and it comes from a toric model.

Proof.

Since the metric is algebraic, there exist a proper $K^{\circ}$ - scheme $\mathcal{X}$ and a line bundle $\mathcal{L}$ on $\mathcal{X}$ such that the base change of $(\mathcal{X},\mathcal{L})$ to $K$ is isomorphic to $(X_{\Sigma},L^{\otimes e})$ . Let $\{{\mathcal{U}}_{i},s_{i}\}$ be a trivialization of $\mathcal{L}$ . Let $C_{i}={\operatorname{red}}^{-1}({\mathcal{U}}_{i}\cap\mathcal{X}_{o})$ . The subsets $C_{i}$ form a finite closed cover of $X_{\Sigma}^{{\text{\rm an}}}$ . On ${\mathcal{U}}_{i}$ we can write $s_{\Psi}^{\otimes e}=g_{i}s_{i}$ for certain rational function $g_{i}$ . Therefore, on $C_{i}$ , we have $\log\|s_{\Psi}(p)\|=\frac{\log|g(p)|}{e}$ . By Lemma 5.48, it follows that there is a finite closed cover of $N_{\mathbb{R}}$ and the restriction of $\psi_{\|\cdot\|}$ to each of these closed subsets is rational piecewise affine. Therefore $\psi_{\|\cdot\|}$ is rational piecewise affine. The second statement follows from the first and Corollary 5.47. ∎

The next point we study is how to turn a non-toric metric into a toric one. Since the image of $\theta_{0}$ consists of fixed points under the action of $\mathbb{S}^{{\text{\rm an}}}$ (see Proposition-Definition 5.2), we may think of it as the analogue, in the non-Archimedean case, of a Haar measure of volume $1$ on the compact torus $\mathbb{S}^{{\text{\rm an}}}$ .

Let $\Psi$ be a virtual support function on $\Sigma$ . Write $L=\mathcal{O}(D_{\Psi})$ and $s=s_{\Psi}$ . Let $\|\cdot\|$ be a metric on $L^{{\text{\rm an}}}$ , non-necessarily toric. Then we define $\psi_{\|\cdot\|}\colon N_{\mathbb{R}}\to\mathbb{R}$ by

(5.50)

\psi_{\|\cdot\|}(u)=\frac{1}{\lambda_{K}}\log\|s(\theta_{0}({\operatorname{\mathbf{e}}}_{K}(u)))\|.

Note that, if $\|\cdot\|$ is a toric metric, the definition of $\psi_{\|\cdot\|}$ we have just given agrees with the one given in §5.2. This is clear because, if the metric is toric, then $\|s(p)\|=\|s(\theta_{0}\rho_{0}(p))\|$ .

Proposition 5.51.

The assignment that, to a local section $s$ of $L$ gives the function defined as $\|s(\theta_{0}\rho_{0}(p)\|$ for $p\in X^{{\text{\rm an}}}_{\Sigma}$ , is a toric metric on $L^{{\text{\rm an}}}$ , that we denote $\|\cdot\|_{\mathbb{S}}$ . Moreover, $\psi_{\|\cdot\|}=\psi_{\|\cdot\|_{\mathbb{S}}}$ .

Proof.

As in the proof of Proposition 5.16, we can verify that the function $\psi_{\|\cdot\|}-\Psi$ can be extended to a continuous function on $N_{\Sigma}$ . Using that $\theta_{0}$ is a section of $\rho_{0}$ and the image of $\theta_{0}$ consists of points which are fixed under the action of $\mathbb{S}^{{\text{\rm an}}}$ , we also verify that $\|\cdot\|_{\mathbb{S}}$ is the toric metric associated to $\psi_{\|\cdot\|}$ by the same proposition. ∎

The relationship between toric algebraic metrics and rational piecewise functions of Theorem 5.49 can be extended to the case when the metric is non-toric.

Proposition 5.52.

Let $\|\cdot\|$ be an algebraic metric. Then the function $\psi_{\|\cdot\|}$ is rational piecewise affine.

Proof.

Just observe that in the proof of Theorem 5.49 one does not use the fact that the metric is toric. ∎

We now study the effect of taking a field extension. Let $K\subset H$ be a finite extension of fields that are complete with respect to an absolute value associated to a nontrivial discrete valuation. We assume that the absolute value of $H$ is an extension of the absolute value of $K$ . Let ${H}^{\circ}$ be the valuation ring of $H$ , ${H}^{\circ\circ}$ the maximal ideal, $\varpi^{\prime}$ a generator of the maximal ideal, $\lambda_{H}=\log(|\varpi^{\prime}|^{-1})$ . Let $e_{H/K}$ be the ramification degree of the extension. Hence $\lambda_{K}=e_{H/K}\lambda_{H}$ .

Proposition 5.53.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$ with $\Sigma=\operatorname{rec}(\Pi)$ .

(1)

Let $X_{\Sigma,K}$ and $X_{\Sigma,H}$ denote the toric varieties defined by $\Sigma$ over $K$ and $H$ respectively. Then

X_{\Sigma,H}=\operatorname{Spec}(H)\times X_{\Sigma,K}.

Moreover there is a commutative diagram

where the horizontal map is induced by the restriction of seminorms.

(2)

Let $\Pi^{\prime}$ be the polyhedral complex in $N_{\mathbb{R}}$ obtained from $\Pi$ by applying a homothety of ratio $e_{H/K}$ . Then

${\mathcal{X}}_{\Pi^{\prime},{H}^{\circ}}=\operatorname{Nor}(\operatorname{Spec}(H^{\circ})\times{\mathcal{X}}_{\Pi,K^{\circ}}),$

where $\operatorname{Nor}$ denotes the normalization of a scheme.
(3)

Let $\psi$ be a rational piecewise linear function on $\Pi$ and denote $\Psi=\operatorname{rec}(\psi)$ . Let $L=\mathcal{O}(D_{\Psi})$ be the line bundle on $X_{\Sigma,K}$ determined by $\Psi$ and let $\|\cdot\|$ be the metric on $L^{{\text{\rm an}}}$ determined by $\psi$ . Let $L^{\prime}$ be the line bundle obtained by base change and $\|\cdot\|^{\prime}$ the metric obtained by inverse image. Then

$\psi_{\|\cdot\|^{\prime}}(u)=(\psi e_{H/K})(u)=e_{H/K}\psi(e_{H/K}^{-1}u).$

(4)

There is a commutative diagram

Proof.

The statement (1) can be checked locally. Let $\sigma$ be a cone of $\Sigma$ . Then

X_{\sigma,H}=\operatorname{Spec}(H[M_{\sigma}])=\operatorname{Spec}(K[M_{\sigma}]\underset{K}{\otimes}H)=\operatorname{Spec}(H)\underset{K}{\times}X_{\sigma,K}.

This proves the first assertion. The commutativity of the diagram follows from the fact that the map $X_{\Sigma,H}^{{\text{\rm an}}}\to X^{{\text{\rm an}}}_{\Sigma,K}$ is given by the restriction of seminorms.

The statement (2) can also be checked locally. Let $\Lambda$ be a polyhedron of $\Pi$ . Let $\Lambda^{\prime}=e_{K^{\prime}/K}\Lambda$ . Then it is clear that

K^{\circ}[{\mathcal{X}}_{\Lambda}]\underset{K^{\circ}}{\otimes}H^{\circ}\subset H^{\circ}[{\mathcal{X}}_{\Lambda^{\prime}}].

Since the right-hand side ring is integrally closed, the integral closure of the left side ring is contained in the right side ring. Therefore we need to prove that $H[{\widetilde{M}}_{\Lambda^{\prime}}]$ is integral over the left side ring. Let $(a,l)\in{\widetilde{M}}_{\Lambda^{\prime}}$ . Thus $(e_{H/K}a,l)\in{\widetilde{M}}_{\Lambda}$ . Then the monomial $\chi^{a}\varpi^{\prime}{}^{l}\in H^{\circ}[{\mathcal{X}}_{\Lambda^{\prime}}]$ satisfies

(\chi^{a}\varpi^{\prime}{}^{l})^{e_{H/K}}=(\chi^{e_{H/K}a}\varpi^{l})\in K^{\circ}[{\mathcal{X}}_{\Lambda}]\underset{K^{\circ}}{\otimes}H^{\circ}.

Hence $\chi^{a}\varpi^{\prime}{}^{l}$ is integral over $K^{\circ}[{\mathcal{X}}_{\Lambda}]\underset{K^{\circ}}{\otimes}H^{\circ}.$ Since these monomials generate $H^{\circ}[{\mathcal{X}}_{\Lambda^{\prime}}]$ , we obtain the result.

To prove (3), let $p^{\prime}\in X^{{\text{\rm an}}}_{0,H}$ and let $p\in X^{{\text{\rm an}}}_{0,K}$ be the corresponding point. Then ${\operatorname{val}}_{K}(p)=\frac{{\operatorname{val}}_{H}(p^{\prime})}{e_{H/K}}$ . Therefore, if we write $u={\operatorname{val}}_{K}(p)$ and $u^{\prime}=\frac{{\operatorname{val}}_{H}(p^{\prime})}{e_{H/K}}$ , we have

\psi_{\|\cdot\|^{\prime}}(u^{\prime})=\frac{1}{\lambda_{H}}\log\|s(p^{\prime})\|^{\prime}=\frac{e_{H/K}}{\lambda_{K}}\log\|s(p)\|=e_{H/K}\psi(u)=e_{H/K}\psi(u^{\prime}/e_{H/K}).

Finally, statement (4) follows directly from the definition of $\theta_{\Sigma}$ because the horizontal arrow is given by the restriction of seminorms. ∎

5.5. The one-dimensional case

We now study in detail the one-dimensional case. Besides being a concrete example of the relationship between functions, models, metrics, and measures, it is also a crucial step in the proof that a toric metric is semipositive if and only if the corresponding function is concave. Of this equivalence, up to now we have proved only one implication and the reverse implication will be proved in the next section.

The only complete one-dimensional toric variety over a field is the projective line. Since, by Proposition 5.53 and Proposition 2.35 we know the effect of taking finite extensions of the field $K$ , we can use the following result to reduce any model of $\mathbb{P}^{1}$ to a simpler form.

Definition 5.54.

Let $K$ be a field complete with respect to an absolute value associated to a nontrivial discrete valuation. Let $K^{\circ}$ be the ring of integers. Let $X$ be a proper curve over $K$ . A semi-stable model of $X$ is a flat proper regular scheme ${\mathcal{X}}$ of finite type over $\operatorname{Spec}(K^{\circ})$ with an isomorphism $X\to{\mathcal{X}}_{\eta}$ , such that the special fibre ${\mathcal{X}}_{o}$ is a reduced normal crossing divisor.

Proposition 5.55.

Let $K$ be a field complete with respect to an absolute value associated to a nontrivial discrete valuation. Let $K^{\circ}$ be the ring of integers. Let ${\mathcal{X}}$ be a proper model over $K^{\circ}$ of $\mathbb{P}^{1}_{K}$ . Then there exists a finite extension $H$ of $K$ with ring of integers ${H}^{\circ}$ , a semi-stable model ${\mathcal{X}}^{\prime}$ of $\mathbb{P}^{1}_{H}$ , and a proper morphism of models $\mathcal{X}^{\prime}\to\mathcal{X}\times\operatorname{Spec}({H}^{\circ})$ .

Proof.

This follows, for instance, from [Liu06, Corollary 2.8]. ∎

Consider the toric variety $X_{\Sigma}\simeq\mathbb{P}^{1}$ . We can choose an isomorphism $N\simeq\mathbb{Z}$ and $N_{\mathbb{R}}\simeq\mathbb{R}$ . Then $\Sigma=\{\mathbb{R}_{-},\{0\},\mathbb{R}_{+}\}$ . Let $0$ denote the invariant point of $\mathbb{P}^{1}_{K}$ corresponding to the cone $\mathbb{R}_{+}$ and $\infty$ the invariant point corresponding to the cone $\mathbb{R}_{-}$ . Let $t$ denote the absolute coordinate of $\mathbb{P}^{1}$ given by the monomial $\chi^{1}$ .

Let $\mathcal{X}$ be a semi-stable model of $\mathbb{P}^{1}_{K}$ . By extending scalars if necessary, we may suppose that all the components of the special fibre are defined over $k=K^{\circ}/K^{\circ\circ}$ and contain a rational point. Since the special fibre $\mathcal{X}_{o}$ is connected and of genus zero, we deduce that the special fibre is a tree of rational curves, each isomorphic to $\mathbb{P}^{1}_{k}$ . Let $D_{0}$ and $D_{\infty}$ denote the horizontal divisors corresponding to the point $0$ and $\infty$ of $\mathbb{P}^{1}_{K}$ . Then, there is a chain of rational curves that links the divisor $D_{0}$ with $D_{\infty}$ that is contained in the special fibre. We will denote the irreducible components of the special fibre that form this chain by $E_{0},\dots,E_{k}$ , in such a way that the component $E_{0}$ meets $D_{0}$ , the component $E_{k}$ meets $D_{\infty}$ and, for $0<i<k$ , the component $E_{i}$ meets only $E_{i-1}$ and $E_{i+1}$ . The other components of $\mathcal{X}_{o}$ will be grouped in branches, each branch has its root in one of the components $E_{i}$ . We will denote by $F_{i,j}$ , $j\in\Theta_{i}$ the components that belong to a branch with root in $E_{i}$ . We are not giving any particular order to the sets $\Theta_{i}$ .

We denote by $E\cdot F$ the intersection product of two $1$ -cycles of ${\mathcal{X}}$ . Since the special fibre is reduced, we have

\operatorname{div}(\varpi)=\sum_{i=0}^{k}\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right).

Again by the assumption of semi-stability, the intersection product of two different components of $\mathcal{X}_{o}$ is either $1$ , if they meet, or zero, if they do not meet. Since the intersection product of $\operatorname{div}(\varpi)$ with any component of $\mathcal{X}_{o}$ is zero, we deduce that, if $E$ is any component of $\mathcal{X}_{o}$ , the self-intersection product $E\cdot E$ is equal to minus the number of components that meet $E$ . In particular, all components $F_{i,j}$ that are terminal, are $(-1)$ -curves. By Castelnuovo Criterion, we can successively blow-down all the components $F_{i,j}$ to obtain a new semi-stable model of $\mathbb{P}^{1}_{K}$ whose special fibre consist of a chain of rational curves. For reasons that will become apparent later we denote this model as ${\mathcal{X}}_{\mathbb{S}}$ .

Lemma 5.56.

If we view $t$ as a rational function on $\mathcal{X}$ , then there is an integer $a$ such that

\operatorname{div}(t)=D_{0}-D_{\infty}+\sum_{i=0}^{k}(a-i)\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right).

Proof.

It is clear that

\operatorname{div}(t)=D_{0}-D_{\infty}+\sum_{i=0}^{k}a_{i}E_{i}+\sum_{j\in\Theta_{i}}a_{i,j}F_{i,j}

for certain coefficients $a_{i}$ and $a_{i,j}$ that we want to determine as much as possible.

If a component $E$ of $\mathcal{X}_{0}$ , with coefficient $a$ , does not meet $D_{0}$ nor $D_{\infty}$ , but meets $r\geq 1$ other components, and the coefficients of $r-1$ of these components are equal to $a$ , while the coefficient of the remaining component is $b$ , we obtain that

0=\operatorname{div}(t)\cdot E=aE\cdot E+a(r-1)+b=-ra+a(r-1)+b=b-a

Thus $b=a$ . Starting with the components $F_{i,j}$ that are terminal, we deduce that, for all $i$ and $j\in\Theta_{i}$ , $a_{i}=a_{i,j}$ . Therefore,

\operatorname{div}(t)=D_{0}-D_{\infty}+\sum_{i=0}^{k}a_{i}\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right).

In particular, the lemma is proved for $k=0$ . Assume now that $k>0$ .

It only remains to show that $a_{i}=a_{0}-i$ , that we prove by induction. For $i=1$ , we compute

0=\operatorname{div}(t)\cdot E_{0}=D_{0}\cdot E_{0}+a_{0}E_{0}\cdot E_{0}+a_{0}\sum_{j\in\Theta_{0}}F_{0,j}\cdot E_{0}+a_{1}E_{1}\cdot E_{0}=1-a_{0}+a_{1}.

Thus $a_{1}=a_{0}-1$ . For $1<i\leq k$ , by induction hypothesis, $a_{i-1}=a_{i-2}-1$ . Then

0=\operatorname{div}(t)\cdot E_{i-1}=a_{i-2}-2a_{i-1}+a_{i}=1-a_{i-1}+a_{i}.

Thus $a_{i}=a_{i-1}-1=a_{0}-i$ , proving the lemma. ∎

The determination of $\operatorname{div}(t)$ allows us to give a partial description of the map ${\operatorname{red}}\colon X_{\Sigma}^{{\text{\rm an}}}\to\mathcal{X}_{o}$ . For us, the most interesting points of $\mathcal{X}_{o}$ are the points $q_{0}:=D_{0}\cap E_{0}$ , $q_{i}:=E_{i-1}\cap E_{i}$ , $i=1,\dots,k$ , $q_{k+1}:=E_{k}\cap D_{\infty}$ and the generic points of the components $E_{i}$ that we denote $\eta_{i}$ , $i=0,\dots,k$ .

Lemma 5.57.

Let $p\in X_{\Sigma}^{{\text{\rm an}}}$ . Then

{\operatorname{red}}(p)=\begin{cases}q_{0},&\text{ if }|t(p)|<|\varpi|^{a}\\ q_{i},\ i=1\dots,k,&\text{ if }|\varpi|^{a-i+1}<|t(p)|<|\varpi|^{a-i}\\ q_{k+1},&\text{ if }|\varpi|^{a-k}<|t(p)|\\ \eta_{i},\ i=0\dots,k,&\text{ if }|t(p)|=|\varpi|^{a-i}\text{ and }p\in\operatorname{im}(\theta_{\Sigma}).\end{cases}

Proof.

Let $1\leq i\leq k$ . The rational function $x:=t\varpi^{-a+i}$ has a zero of order one along the component $E_{i-1}$ and the support of its divisor does not contain the component $E_{i}$ . On the other hand, the rational function $y:=t^{-1}\varpi^{a-i+1}$ has a zero of order one along the component $E_{i}$ and the support of its divisor does not contain the component $E_{i-1}$ . Thus $\{x,y\}$ is a system of parameters in a neighbourhood of $q_{i}$ . We denote

A=K^{\circ}[t\varpi^{-a+i},t^{-1}\varpi^{a-i+1}]\simeq K^{\circ}[x,y]/(xy-\varpi).

The local ring at the point $q_{i}$ is $A_{(x,y)}$ . Let $p$ be a point such that $|\varpi|^{a-i+1}<|t(p)|<|\varpi|^{a-i}$ . Therefore, for $f\in A$ we have $|f(p)|\leq 1$ . Moreover, if $f\in(x,y)$ , then $|f(p)|<1$ . Since the ideal $(x,y)$ is maximal, we deduce that, for $f\in A$ , the condition $|f(p)|<1$ is equivalent to the condition $f\in(x,y)$ . This implies that ${\operatorname{red}}(p)=q_{i}$ . A similar argument works for $q_{0}$ and $q_{k+1}$ .

Assume now that $p\in\operatorname{im}(\theta_{\Sigma})$ and that $|t(p)|=|\varpi|^{a-i}$ . If $i\not=0$ we consider again the ring $A$ , but in this case $|x(p)|=|t(p)\varpi^{-a+i}|=1$ . Let $I=\{f\in A\mid|f(p)|<1\}$ . It is clear that $(y,\varpi)\subset I$ . For $f=\sum_{m\in\mathbb{Z}}\beta_{m}t^{m}\in A$ , since $p\in\operatorname{im}(\theta_{\Sigma})$ , we have

|f(p)|=\sup_{m}(|\beta_{m}||t(p)|^{m}).

This implies that $I\subset(y,\varpi)$ . Hence $I$ is the ideal that defines the component $E_{i}$ and this is equivalent to ${\operatorname{red}}(p)=\eta_{i}$ . The case $i=0$ is analogous. ∎

The image by ${\operatorname{red}}$ of the remaining points of $X_{\Sigma}^{{\text{\rm an}}}$ is not characterized only by the value of $|t(p)|$ . Using a proof similar to that of the lemma, one can show that, if $|t(p)|=|\varpi|^{a-i}$ then ${\operatorname{red}}(p)$ belongs either to $E_{i}$ or to any of the components $F_{i,j}$ , $j\in\Theta_{i}$ .

We denote by $\xi_{i}$ (resp. $\xi_{i,j}$ ) the point of $X_{\Sigma}^{{\text{\rm an}}}$ corresponding to the component $E_{i}$ (resp. $F_{i,j}$ ). That is, ${\operatorname{red}}(\xi_{i})=\eta_{i}$ and ${\operatorname{red}}(\xi_{i,j})=\eta_{i,j}$ , where $\eta_{i,j}$ is the generic point of $F_{i,j}$ (see (2.15) and (2.14)).

Lemma 5.58.

Let $0\leq i\leq k$ . Then, for every $j\in\Theta_{i}$ ,

{\operatorname{val}}_{K}(\xi_{i})={\operatorname{val}}_{K}(\xi_{i,j})=a-i,

where $a$ is the integer of Lemma 5.56.

Proof.

We consider the rational function $\varpi^{-a+i}t$ . Since the support of $\operatorname{div}(\varpi^{-a+i}t)$ does not contain the component $E_{i}$ nor any of the components $F_{i,j}$ , we have that

|\varpi^{-a+i}t(\xi_{i})|=|\varpi^{-a+i}t(\xi_{i,j})|=1.

Since $t=\chi^{1}$ , we deduce, using equation (5.4), that

{\operatorname{val}}_{K}(\xi_{i})=\frac{-\log|\chi^{1}(x_{i})|}{\lambda_{K}}=\frac{-\log|\varpi^{a-i}|}{-\log|\varpi|}=a-i.

∎

Let now $\Psi$ be a virtual support function on $\Sigma$ . It can be written as

\Psi(u)=\begin{cases}m_{\infty}u,&\text{ if }u\leq 0,\\ m_{0}u,&\text{ if }u\geq 0.\end{cases}

for some $m_{0},m_{\infty}\in\mathbb{Z}$ . Then, $L=\mathcal{O}(D_{\Psi})\simeq\mathcal{O}(m_{\infty}-m_{0})$ , and $\operatorname{div}(s_{\Psi})=-m_{0}[0]+m_{\infty}[\infty]$ . Let $\mathcal{L}$ be a model over $\mathcal{X}$ of $L^{\otimes e}$ . If we consider $s_{\Psi}^{\otimes e}$ as a rational section of $\mathcal{L}$ , then

(5.59)

\operatorname{div}(s_{\Psi}^{\otimes e})=-em_{0}D_{0}+em_{\infty}D_{\infty}+\sum_{i=0}^{k}\left(\alpha_{i}E_{i}+\sum_{j\in\Theta_{i}}\alpha_{i,j}F_{i,j}\right)

for certain coefficients $\alpha_{i}$ and $\alpha_{i,j}$ . Let $\|\cdot\|$ be the metric on $L^{{\text{\rm an}}}$ determined by this model.

Lemma 5.60.

The function $\psi_{\|\cdot\|}$ is given by

\psi_{\|\cdot\|}(u)=\begin{cases}m_{0}u-m_{0}a-\frac{\alpha_{0}}{e},&\text{ if }u\geq a,\\ \frac{(\alpha_{i+1}-\alpha_{i})u-(\alpha_{i+1}-\alpha_{i})(a-i)-\alpha_{i}}{e},&\text{ if }a-i\geq u\geq a-i-1,\\ m_{\infty}u-m_{\infty}(a-k)-\frac{\alpha_{k}}{e},&\text{ if }a-k\geq u.\end{cases}

In other words, if $\Pi$ is the polyhedral complex in $N_{\mathbb{R}}$ given by the intervals

(-\infty,a-k],\quad[a-i,a-i+1],\ i=1,\dots,k,\quad[a,\infty),

then $\psi_{\|\cdot\|}$ is the rational piecewise affine function on $\Pi$ characterized by the conditions

(1)

$\operatorname{rec}(\psi_{\|\cdot\|})=\Psi$ ,
(2)

the value of $\psi_{\|\cdot\|}$ at the point $a-i$ is $-\alpha_{i}/e$ .

Proof.

Let $p\in\operatorname{im}\theta_{\Sigma}$ be such that ${\operatorname{val}}_{K}(p)>a$ , hence $|t(p)|<|\varpi|^{a}$ . By Lemma 5.57, this implies that ${\operatorname{red}}(p)=q_{0}$ . In a neighbourhood of $q_{0}$ , the divisor of the rational section $s_{\Psi}^{\otimes e}t^{em_{0}}\varpi^{-\alpha_{0}-em_{0}a}$ is zero, and so

\|s_{\Psi}^{\otimes e}(p)t^{em_{0}}(p)\varpi^{-\alpha_{0}-em_{0}a}\|=1.

Set $u={\operatorname{val}}(p)$ . Then,

	$\displaystyle\psi_{\\|\cdot\\|}(u)$	$\displaystyle=\frac{\log\\|s_{\Psi}^{\otimes e}(p)\\|}{e\lambda_{K}}$
		$\displaystyle=\frac{-em_{0}\log\|t(p)\|+(\alpha_{0}+em_{0}a)\log\|\varpi\|}{-e\log\|\varpi\|}$
		$\displaystyle=m_{0}(u-a)-\frac{\alpha_{0}}{e}.$

The other cases are proved in a similar way. ∎

Since $\operatorname{rec}(\Pi)=\Sigma$ , this polyhedral complex defines a toric model $\mathcal{X}_{\Pi}$ of $X_{\Sigma}$ .

Proposition 5.61.

The identity map of $X_{\Sigma}$ extend to an isomorphism of models $\mathcal{X}_{\mathbb{S}}\to\mathcal{X}_{\Pi}$ .

Proof.

The special fibre of $\mathcal{X}_{\Pi}$ is a chain of rational curves $E_{i}$ , $i=0,\dots,k$ , corresponding to the points $a-i$ . The monomial $\chi^{1}$ is a section of the trivial line bundle and corresponds to the function $\psi(u)=-u$ . Using Proposition 4.84 we obtain that

\operatorname{div}(\chi^{1})=D_{0}-D_{\infty}+\sum_{i=0}^{k}(a-i)E_{i},

where $D_{0}$ and $D_{\infty}$ are again the horizontal divisors determined by the points $0$ and $\infty$ .

Since the vertices of the polyhedral complex $\Pi$ are integral, by equation (4.87), we deduce that $\operatorname{div}(\varpi)$ is reduced.

Then the result follows from [Lic68, Corollary 1.13] using an explicit description of the local rings at the points of the special fibre as in the proof of Lemma 5.57. ∎

From Proposition 5.61 we obtain a proper morphism $\pi\colon\mathcal{X}\to\mathcal{X}_{\Pi}$ . On $\mathcal{X}$ we had a line bundle $\mathcal{L}$ and $s_{\Psi}^{\otimes e}$ was considered as a rational section of this line bundle. Let $D=\operatorname{div}(s_{\Psi}^{\otimes e})$ be the divisor given by equation (5.59). We denote

(5.62)

D_{\mathbb{S}}=\pi_{\ast}D=-em_{0}D_{0}+em_{\infty}D_{\infty}+\sum_{i=0}^{k}\alpha_{i}E_{i}.

By Proposition 4.84 and Lemma 5.60 we see that $D_{\mathbb{S}}=D_{e\psi_{h}}$ . Thus $\mathcal{O}(D_{\mathbb{S}})$ is a toric model of $L^{\otimes e}$ . Recall that $\|\cdot\|$ denoted the metric associated to the model $\mathcal{O}(D)$ . Let $\|\cdot\|_{\mathbb{S}}$ be the toric metric obtained from $\|\cdot\|$ as in Proposition 5.51. By this proposition and equation (5.62), the metric $\|\cdot\|_{\mathbb{S}}$ agrees with the metric defined by the model $\mathcal{O}(D_{\mathbb{S}})$ . Thus, we have identified a toric model that corresponds to the metric $\|\cdot\|_{\mathbb{S}}$ . This allows us to compute directly the associated measure.

Proposition 5.63.

Let $X_{\Sigma}\simeq\mathbb{P}^{1}_{K}$ be a one-dimensional toric variety over $K$ . Let $L\simeq\mathcal{O}(D_{\Psi})$ be a toric line bundle and let $\|\cdot\|$ be an algebraic metric defined by a semi-stable model and let $\|\cdot\|_{\mathbb{S}}$ be the associated toric metric. Then

c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\left(c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}\right).

Proof.

Since the special fibre is reduced, by equation (2.29)

c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}=\frac{1}{e}\sum_{i=0}^{k}\left(\deg_{\mathcal{L}}E_{i}\delta_{\xi_{i}}+\sum_{j\in\Theta_{i}}\deg_{\mathcal{L}}F_{i,j}\delta_{\xi_{i,j}}\right).

Denote this measure temporarily by $\mu$ . Then

	$\displaystyle(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\mu$	$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\left(\deg_{\mathcal{L}}E_{i}+\sum_{j\in\Theta_{i}}\deg_{\mathcal{L}}F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\left(D\cdot E_{i}+\sum_{j\in\Theta_{i}}D\cdot F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\sum_{l=0}^{k}\left(\alpha_{l}E_{l}+\sum_{s\in\Theta_{l}}\alpha_{l,s}F_{l,s}\right)\cdot\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\left(\alpha_{i-1}E_{i-1}+\alpha_{i}E_{i}+\alpha_{i+1}E_{i+1}\right)\cdot\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}(\alpha_{i-1}-2\alpha_{i}+\alpha_{i+1})\delta_{\xi_{i}}.$

In the previous computation, we have used that, since $E_{l}\cdot\operatorname{div}(\varpi)=F_{l,s}\cdot\operatorname{div}(\varpi)=0$ , then

	$\displaystyle F_{l,s}\cdot(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j})$	$\displaystyle=0,\text{ for all }i,j,l,s,$
	$\displaystyle E_{l}\cdot(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j})$	$\displaystyle=\begin{cases}0,&\text{ if }l\not=i-1,i,i+1,\\ 1,&\text{ if }l=i-1,i+1,\\ -2,&\text{ if }l=i.\end{cases}$

An analogous computation shows that

(5.64)

c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}=\frac{1}{e}\sum_{i=0}^{k}(\alpha_{i-1}-2\alpha_{i}+\alpha_{i+1})\delta_{\xi_{i}}.

∎

Using Proposition 5.55 we can extend the above result to the case when the model is not semi-stable.

Corollary 5.65.

Let $X_{\Sigma}\simeq\mathbb{P}^{1}_{K}$ be a one-dimensional toric variety over $K$ . Let $L\simeq\mathcal{O}(D_{\Psi})$ be a toric line bundle, $\|\cdot\|$ an algebraic metric, and $\|\cdot\|_{\mathbb{S}}$ the associated toric metric. Then

c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\left(c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}\right).

Proof.

Let $(\mathcal{X},\mathcal{L})$ be a model of $(X_{\Sigma},L^{\otimes e})$ that realizes the algebraic metric $\|\cdot\|$ . For short, denote $\mu=c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}$ and $\mu_{\mathbb{S}}=c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ . By Proposition 5.55 there is a non-Archimedean field $H$ over $K$ and a semi-stable model ${\mathcal{X}}^{\prime}$ of $X_{\Sigma,H}$ . We may further assume that all the components of the special fibre of ${\mathcal{X}}^{\prime}$ are defined over $H^{\circ}/H^{\circ\circ}$ . Let $(L^{\prime},\|\cdot\|^{\prime})$ be the metrized line bundle obtained by base change to $H$ . Then $(\|\cdot\|^{\prime})_{\mathbb{S}}$ is obtained from $\|\cdot\|_{\mathbb{S}}$ by base change. We denote by $\pi\colon X^{{\text{\rm an}}}_{\Sigma,H}\to X^{{\text{\rm an}}}_{\Sigma,K}$ the map of analytic spaces. Be will denote by $\mu^{\prime}$ , $\mu^{\prime}_{\mathbb{S}}$ , $\theta_{\Sigma}^{\prime}$ and $\rho_{\Sigma}^{\prime}$ the corresponding objects for $X_{\Sigma,H}$ . Then, by Proposition 2.35 and Proposition 5.53,

\mu_{\mathbb{S}}=\pi_{\ast}\mu^{\prime}_{\mathbb{S}}=\pi_{\ast}(\theta^{\prime}_{\Sigma})_{\ast}(\rho^{\prime}_{\Sigma})_{\ast}\mu^{\prime}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\pi_{\ast}\mu^{\prime}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\mu.

∎

We can now relate semipositivity of the metric with concavity of the associated function on the one-dimensional case.

Corollary 5.66.

Let $X_{\Sigma}\simeq\mathbb{P}^{1}_{K}$ be a one-dimensional toric variety over $K$ . Let $(L,s)$ be a toric line bundle with a toric section and let $\|\cdot\|$ be a semipositive algebraic metric. Then $\|\cdot\|_{\mathbb{S}}$ is a semipositive toric algebraic metric and $\psi_{\|\cdot\|}$ is concave.

Proof.

Since $\|\cdot\|$ is semipositive, $c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}$ is a positive measure. By Corollary 5.65, $c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ is a positive measure. Hence $\|\cdot\|_{\mathbb{S}}$ is a semipositive toric metric. By equation (5.64) and Lemma 5.60, the positivity of $c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ implies that the function $\psi_{\|\cdot\|}=\psi_{\|\cdot\|_{\mathbb{S}}}$ is concave. ∎

5.6. Algebraic metrics and their associated measures

We come back to the case of general dimension. Let $\Sigma$ be a complete fan, $\Psi$ a support function on $\Sigma$ and $(L,s)=(L_{\Psi},s_{\Psi})$ . Since $\Psi$ is a support function, the line bundle $L$ is generated by global sections.

Proposition 5.67.

Let $\|\cdot\|$ be a semipositive algebraic metric on $L^{{\text{\rm an}}}$ . Then the function $\psi_{\|\cdot\|}$ is concave.

Proof.

Assume that $\|\cdot\|$ is semipositive. Let $u_{0}$ be a point of $N_{\mathbb{Q}}$ and let $v_{0}\in N$ be primitive. Since the condition of being concave is closed, if we prove that, for all choices of $u_{0}\in N_{\mathbb{Q}}$ and $v_{0}\in N$ , the restriction of $\psi_{\|\cdot\|}$ to the line $u_{0}+\mathbb{R}v_{0}$ is concave, we will deduce that the function $\psi_{\|\cdot\|}$ is concave. Let $e\in\mathbb{N}^{\times}$ such that $eu_{0}\in N$ . Then $H=K(\varpi^{1/e})$ is a finite extension of $K$ and there is a unique extension of the absolute value of $K$ to $H$ . We will denote with ^′ the objects obtained by base change to $H$ . Let $p\in X_{0,H}(H)$ such that ${\operatorname{val}}_{H}(p)=eu_{0}$ . We consider the affine map $A\colon\mathbb{Z}\to N$ given by $l\mapsto v_{0}l+eu_{0}$ , and let $H$ be the linear part of $A$ . We consider the equivariant morphism $\varphi=\varphi_{p,H}\colon\mathbb{P}^{1}_{H}\to X_{\Sigma,H}$ of Theorem 4.9. The metric $\|\cdot\|$ induces an algebraic semipositive metric $\varphi^{\ast}\|\cdot\|^{\prime}$ on the restriction of $L^{\prime}$ (the line bundle obtained from $L$ by base change to $H$ ) to $\mathbb{P}^{1}_{H}$ . By propositions 5.24 and 5.53(3) we obtain that

\psi_{\varphi^{\ast}\|\cdot\|}(u)=e\psi_{\|\cdot\|}(u_{0}+e^{-1}uv_{0}).

By Corollary 5.66 the left-hand side function is concave. Thus the restriction of $\psi_{\|\cdot\|}$ to $u_{0}+\mathbb{R}v_{0}$ is concave. We conclude that $\psi=\psi_{\|\cdot\|}$ is concave. ∎

Corollary 5.68.

Let $\|\cdot\|$ be a semipositive algebraic metric on $L^{{\text{\rm an}}}$ . Then the toric metric $\|\cdot\|_{\mathbb{S}}$ is a semipositive toric algebraic metric.

Proof.

By Proposition 5.67, the function $\psi_{\|\cdot\|}$ is concave. By Proposition 5.52 it is also rational piecewise affine. By Corollary 5.47, the metric $\|\cdot\|_{\mathbb{S}}=\|\cdot\|_{\psi}$ is toric algebraic and semipositive. ∎

Putting together Proposition 5.67 and Theorem 5.49, we see that the relationship between semipositivity of the metric and concavity of the associated function given in the Archimedean case by Proposition 5.29 carries over to the non-Archimedean case.

Corollary 5.69.

Let $\|\cdot\|$ be a toric algebraic metric and $\psi_{\|\cdot\|}$ the associated function. Then the metric is semipositive if and only if the function $\psi_{\|\cdot\|}$ is concave.

We can now characterize the Chambert-Loir measure associated to a toric semipositive algebraic metric.

Theorem 5.70.

Let $\|\cdot\|$ be a toric semipositive algebraic metric on $L^{{\text{\rm an}}}$ and let $\psi=\psi_{\|\cdot\|}$ be the associated function on $N_{\mathbb{R}}$ . Let $c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}$ be the associated measure. Then

(5.71)

({\operatorname{val}}_{K})_{\ast}(c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}})=n!{\overline{\mathcal{M}}}_{M}(\psi),

where ${\overline{\mathcal{M}}}(\psi)$ is the measure of Definition 5.32. Moreover,

(5.72)

c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}({\operatorname{\mathbf{e}}}_{K})_{\ast}n!{\overline{\mathcal{M}}}_{M}(\psi).

Proof.

Since the metric is semipositive and toric, by Proposition 5.67 the function $\psi$ is concave. Since, moreover it is algebraic, by Theorem 5.49 it is defined by a toric model $({\mathcal{X}}_{\Pi},D_{\psi},e)$ of $(X_{\Sigma},D_{\Psi})$ in the equivalence class determined by $\psi$ . As in Remark 4.66, the irreducible components of ${\mathcal{X}}_{\Pi,o}$ are in bijection with the vertices of $\Pi$ . For each vertex $v\in\Pi^{0}$ , let $\xi_{v}$ be the point of $X_{\Sigma}^{{\text{\rm an}}}$ corresponding to the generic point of $V(v)$ defined by equation (2.15). Then, by equation (2.29),

c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}=\frac{1}{e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{\xi_{v}}.

Thus, by Corollary 5.40,

({\operatorname{val}}_{K})_{\ast}(c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}})=\frac{1}{e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{v}.

But, using Proposition 3.95 and Proposition 4.105, the Monge-Ampère measure is given by

	$\displaystyle\mathcal{M}_{M}(\psi)$	$\displaystyle=\frac{1}{e^{n}}\mathcal{M}_{M}(e\psi)$
		$\displaystyle=\frac{1}{e^{n}}\sum_{v\in\Pi^{0}}\operatorname{vol}_{M}(v^{\ast})\delta_{v}$
		$\displaystyle=\frac{1}{n!e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{v}.$

Since $\mathcal{M}_{M}(\psi)$ is a finite sum of Dirac deltas, we obtain that

{\overline{\mathcal{M}}}_{M}(\psi)=\frac{1}{n!e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{v}.

Hence we have proved (5.71). To prove equation (5.72) we just observe that $x_{v}=(\theta_{0}\circ{\operatorname{\mathbf{e}}}_{K})(v)$ . ∎

5.7. Approachable and integrable metrics

We are now in position to characterize the approachable metrics. In this section $K$ is either $\mathbb{R}$ , $\mathbb{C}$ or a complete field with respect to an absolute value associated to a nontrivial discrete valuation. We fix a complete fan $\Sigma$ of $N_{\mathbb{R}}$ , so that $X_{\Sigma}$ is proper. Let $\Psi$ be a support function on $\Sigma$ , $\Delta_{\Psi}$ the corresponding polytope, and $(L_{\Psi},s_{\Psi})$ the corresponding toric line bundle and section. For short, write $X=X_{\Sigma}$ , $L=L_{\Psi}$ and $s=s_{\Psi}$ .

Theorem 5.73.

Assume the previous hypothesis.

(1)

The assignment $\|\cdot\|\mapsto\psi_{\|\cdot\|}$ is a bijection between the space of approachable toric metrics on $L^{{\text{\rm an}}}$ and the space of continuous concave functions $\psi$ on $N_{\mathbb{R}}$ such that $|\psi-\Psi|$ is bounded.
(2)

The assignment $\|\cdot\|\mapsto\psi_{\|\cdot\|}^{\vee}$ is a bijection between the space of approachable toric metrics on $L^{{\text{\rm an}}}$ and the space of continuous concave functions on $\Delta_{\Psi}$ .

Proof.

By Proposition 3.77(2) and Proposition 3.80, the statements (1) and (2) are equivalent.

Let $\|\cdot\|$ be an approachable toric metric. By Corollary 5.17 the function $|\psi_{\|\cdot\|}-\Psi|$ is bounded. By approachability there is a sequence $\|\cdot\|_{l}$ of smooth (resp. algebraic) semipositive metrics that converges to $\|\cdot\|$ . Since $\|\cdot\|$ is toric, $\|\cdot\|_{\mathbb{S}}=\|\cdot\|$ . Hence, the sequence of toric metrics $(\|\cdot\|_{l})_{\mathbb{S}}$ also converges to $\|\cdot\|$ . We denote $\psi_{l}=\psi_{(\|\cdot\|_{l})_{\mathbb{S}}}$ . By Proposition 5.38 and Proposition 5.67 the functions $\psi_{l}$ are concave. Since the sequence $(\psi_{l})_{l}$ converge uniformly to $\psi_{\|\cdot\|}$ , the latter is concave.

Let now $\psi$ be a concave function on $N_{\mathbb{R}}$ such that $|\Psi-\psi|$ is bounded. Then $\psi$ determines a metric $\|\cdot\|$ on the restriction of $L^{{\text{\rm an}}}$ to $X_{0}^{{\text{\rm an}}}$ . Since $\operatorname{stab}(\psi)=\operatorname{stab}(\Psi)=\Delta_{\Psi}$ , by Proposition 3.81 there is a sequence of rational piecewise affine concave functions $\psi_{l}$ that converge uniformly to $\psi$ and with $\operatorname{rec}(\psi_{l})=\Psi$ . By Remark 5.46, the functions $\Psi-\psi_{l}$ can be extended to continuous functions on $N_{\Sigma}$ . Therefore, $\Psi-\psi$ can be extended to a continuous function on $N_{\Sigma}$ . Consequently the metric $\|\cdot\|$ can be extended to $X^{{\text{\rm an}}}$ . Let $\|\cdot\|_{l}$ be the metric associated to $\psi_{l}$ . Then the sequence of metrics $\|\cdot\|_{l}$ converges to $\|\cdot\|$ . By Corollary 5.28, the metrics $\|\cdot\|_{l}$ are approachable. We deduce that $\|\cdot\|$ is approachable. ∎

Remark 5.74.

For the case $K=\mathbb{C}$ , statement (2) in the above result is related to the Guillemin-Abreu classification of Kähler structures on symplectic toric varieties as explained in [Abr03]. By definition, a symplectic toric variety is a compact symplectic manifold of dimension $2n$ together with a Hamiltonian action of the compact torus $\mathbb{S}^{{\text{\rm an}}}\simeq(S^{1})^{n}$ . These spaces are classified by Delzant polytopes of $M_{\mathbb{R}}$ , see for instance [Gui95]. For a given Delzant polytope $\Delta\subset M_{\mathbb{R}}$ , the possible $(S^{1})^{n}$ -invariant Kähler forms on the symplectic toric variety corresponding to $\Delta$ are classified by smooth convex functions on $\Delta^{\circ}$ satisfying some conditions near the border of $\Delta$ . Several differential geometric invariants of a Kähler toric variety can be translated and studied in terms of this convex function, also called the ‘‘symplectic potential’’.

For a smooth positive toric metric $\|\cdot\|$ on $L_{\Psi_{\Delta}}(\mathbb{C})$ , the Chern form defines a Kähler structure on the complex toric variety $X_{\Sigma_{\Delta}}(\mathbb{C})$ . It turns out that the corresponding symplectic potential coincides with minus the function $\psi^{\vee}_{\|\cdot\|}$ . It would be most interesting to explore further this connection.

We now study the compatibility of the restriction of approachable toric metrics to toric orbits and its inverse image by equivariant maps with direct and inverse image of concave functions. This is an extension of propositions 4.99 and 4.108. We start with the case of orbits, and we state a variant of Proposition 5.22 for approachable metrics.

Proposition 5.75.

Let $\|\cdot\|$ be an approachable toric metric on $L^{{\text{\rm an}}}$ , and denote ${\overline{L}}=(L,\|\cdot\|)$ and $\psi=\psi_{{\overline{L}},s}$ the associated concave function on $N_{\mathbb{R}}$ . Let $\sigma\in\Sigma$ and $m_{\sigma}\in M$ such that $\Psi|_{\sigma}=m_{\sigma}|_{\sigma}$ . Let $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ be the projection, $\pi^{\vee}_{\sigma}\colon M(\sigma)_{\mathbb{R}}\to M_{\mathbb{R}}$ the dual inclusion and $\iota\colon V(\sigma)\to X$ the closed immersion. Set $s^{\prime}=\chi^{m_{\sigma}}s$ . Then

(5.76)

\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}=(\pi_{\sigma})_{\ast}(\psi-m_{\sigma}).

Dually, we have that

(5.77)

\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}^{\vee}=(\pi^{\vee}_{\sigma}+m_{\sigma})^{\ast}\psi^{\vee}.

In other words, the Legendre-Fenchel dual of $\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}$ is the restriction of $\psi^{\vee}$ to the face $F_{\sigma}$ translated by $-m_{\sigma}$ .

Proof.

As in the proof of Proposition 4.99, it is enough to prove equation (5.76). By replacing $\psi$ by $\psi-m_{\sigma}$ , we can assume without loss of generality that $m_{\sigma}=0$ . By the continuity of the metric, the function $\psi$ can be extended to a continuous function ${\overline{\psi}}_{\sigma}$ on $N_{\sigma}$ . Fix $u_{0}\in N(\sigma)_{\mathbb{R}}$ , write $s={\overline{\psi}}_{\sigma}(u_{0})$ and let $u\in N_{\mathbb{R}}$ such that $\pi_{\sigma}(u)=u_{0}$ . By definition

(\pi_{\sigma})_{\ast}(\psi)(u_{0})=\sup_{p\in\mathbb{R}\sigma}\psi(u+p).

It is clear that $\sup_{p\in\mathbb{R}\sigma}\psi(u+p)\geq s$ . Suppose that $\sup_{p\in\mathbb{R}\sigma}\psi(u+p)>s$ . Let $q\in\mathbb{R}\sigma$ such that $\psi(u+q)>s$ and let $\varepsilon=(\psi(u+q)-s)/2$ . By the definition of the topology of $N_{\sigma}$ , there exists a $p\in\mathbb{R}\sigma$ such that

(5.78)

s-\varepsilon<\psi(u+p+\sigma)<s+\varepsilon.

Since $\sigma$ is a cone of maximal dimension in $\mathbb{R}\sigma$ , there exists a point $r\in(q+\sigma)\cap(p+\sigma)$ . By the right inequality of equation (5.78) $\psi(u+r)<\psi(u+q)$ . By concavity of $\psi$ this implies that

(5.79)

\lim_{\lambda\to\infty}\psi(u+r+\lambda(r-q))=-\infty.

Since, by construction $u+r+\mathbb{R}_{\geq 0}(r-q)$ is contained in $u+p+\sigma$ , equation (5.79) contradicts the left inequality of equation (5.78). Hence $\sup_{p\in\mathbb{R}\sigma}\psi(u+p)=s$ , which proves equation (5.76). ∎

We now interpret the inverse image of an approachable toric metric by an equivariant map whose image intersects the principal open subset in terms of direct and inverse images of concave functions.

Proposition 5.80.

Let $N_{1}$ and $N_{2}$ be lattices and $\Sigma_{i}$ a complete fan in $N_{i,\mathbb{R}}$ , $i=1,2$ . Let $H\colon N_{1}\to N_{2}$ be a linear map such that, for each $\sigma_{1}\in\Sigma_{1}$ , there exists $\sigma_{2}\in\Sigma_{2}$ with $H(\sigma_{1})\subset\sigma_{2}$ . Let $p\in X_{\Sigma_{2},0}(K)$ and write $A\colon N_{1,\mathbb{R}}\to N_{2,\mathbb{R}}$ for the affine map $A=H+{\operatorname{val}}(p)$ . Let $\|\cdot\|$ be an approachable toric metric on ${\mathcal{O}}(D_{\Psi_{2}})^{{\text{\rm an}}}$ . Then

\psi_{\varphi_{p.H}^{\ast}\|\cdot\|}=A^{*}\psi_{\|\cdot\|}.

Moreover, the Legendre-Fenchel dual of this function is given by

\psi_{\varphi_{p.H}^{\ast}\|\cdot\|}^{\vee}=(H^{\vee})_{*}\big(\psi^{\vee}_{\|\cdot\|}-{\operatorname{val}}(p)\big).

Proof.

The first statement is a direct consequence of Proposition 5.24 while the second one follows from Proposition 3.78(1). ∎

We next characterize the measures associated to an approachable metric.

Theorem 5.81.

Let $\Sigma$ be a complete fan of $N_{\mathbb{R}}$ , let $\Psi$ be a support function on $\Sigma$ and let $L=\mathcal{O}(D_{\Psi})$ . Let $\|\cdot\|$ be an approachable metric on $L^{{\text{\rm an}}}$ and let $\psi=\psi_{\|\cdot\|}$ be the corresponding concave function. Then

(5.82)

({\operatorname{val}}_{K})_{\ast}(c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}})=n!{\overline{\mathcal{M}}}_{M}(\psi).

Moreover, the measure $c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}$ is characterized, in the Archimedean case, by equation (5.82) and the fact of being toric, while in the non-Archimedean case it is given by

c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}({\operatorname{\mathbf{e}}}_{K})_{\ast}n!{\overline{\mathcal{M}}}_{M}(\psi).

Proof.

For short, denote $\mu=({\operatorname{val}}_{K})_{\ast}(c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}})$ . Let $\|\cdot\|_{l}$ be a sequence of semipositive smooth (respectively algebraic) metrics converging to $\|\cdot\|$ . By Proposition 2.33, the measures $c_{1}(L,\|\cdot\|_{l})^{n}\land\delta_{X_{\Sigma}}$ converge to $c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}$ . Therefore, the measures $({\operatorname{val}}_{K})_{\ast}(c_{1}(L,\|\cdot\|_{l})^{n}\land\delta_{X_{\Sigma}})$ converge to the measure $\mu$ on $N_{\Sigma}$ . Proposition 2.37 implies that the measure of $X_{\Sigma}^{{\text{\rm an}}}\setminus X_{0}^{{\text{\rm an}}}$ with respect to $c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}$ is zero. Therefore $N_{\Sigma}\setminus N_{\mathbb{R}}$ has $\mu$ -measure zero. Denote $\psi_{l}=\psi_{(\|\cdot\|_{l})_{\mathbb{S}}}$ . By Proposition 3.108, the measures $\mathcal{M}_{M}(\psi_{l})$ converge to the measure $\mathcal{M}_{M}(\psi)$ . Thus $\mu|_{N_{\mathbb{R}}}=n!\mathcal{M}_{M}(\psi)$ . If we add to this that the measure of $N_{\Sigma}\setminus N_{\mathbb{R}}$ is zero, we deduce equation (5.82). The last statement of the theorem is clear from Theorem 5.33 and Theorem 5.70. ∎

We end this section by characterizing integrable metrics.

Corollary 5.83.

Let $\Sigma$ be a complete fan. Then the map $\|\cdot\|\mapsto\psi_{\|\cdot\|}$ is a bijection between the space of integrable toric metrics on $\mathcal{O}(D_{\Psi})^{{\text{\rm an}}}$ and the space of functions $\psi\in{\overline{\mathscr{D}}}(N_{\mathbb{R}})$ such that $\operatorname{rec}(\psi)=\Psi$ , were ${\overline{\mathscr{D}}}(N_{\mathbb{R}})$ is the space of functions of Definition 3.82.

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. ∎

	$\displaystyle\psi_{\\|\cdot\\|_{{\mathcal{L}}}}(u)$	$\displaystyle=\frac{1}{\lambda_{K}}\log(\\|s(p)\\|_{{\mathcal{L}}})$
		$\displaystyle=\frac{1}{e\lambda_{K}}\log(\|\chi^{-m_{\Lambda}}(p)\varpi^{-l_{\Lambda}}\|)$
		$\displaystyle=\frac{1}{e}(\langle m_{\Lambda},u\rangle+l_{\Lambda})$
		$\displaystyle=\psi(u).$

	$\displaystyle\psi_{\\|\cdot\\|}(u)$	$\displaystyle=\frac{\log\\|s_{\Psi}^{\otimes e}(p)\\|}{e\lambda_{K}}$
		$\displaystyle=\frac{-em_{0}\log\|t(p)\|+(\alpha_{0}+em_{0}a)\log\|\varpi\|}{-e\log\|\varpi\|}$
		$\displaystyle=m_{0}(u-a)-\frac{\alpha_{0}}{e}.$

5. Metrics and measures on toric varieties [02ST]

5. Metrics and measures on toric varieties

5.1. The variety with corners XΣ​(ℝ≥0)X_{\Sigma}(\mathbb{R}_{\geq 0})

Lemma 5.1.

Proof.

Proposition-Definition 5.2.

Proof.

Remark 5.8.

Proposition 5.9.

Proposition 5.10.

5.2. Toric metrics

Proposition 5.11.

Proof.

Definition 5.12.

Definition 5.14.

Proposition 5.16.

Proof.

Corollary 5.17.

Proof.

Example 5.18.

Proposition 5.19.

Proof.

Proposition-Definition 5.20.

Proof.

Proposition 5.21.

Proof.

Proposition 5.22.

Proof.

Corollary 5.23.

Proof.

Proposition 5.24.

Corollary 5.25.

Example 5.26.

Proposition 5.27.

Proof.

Corollary 5.28.

Proof.

5.3. Smooth metrics and their associated measures

Proposition 5.29.

Proof.

Definition 5.32.

Theorem 5.33.

Proof.

Proposition 5.38.

Proof.

5.4. Algebraic metrics from toric models

Lemma 5.39.

Proof.

Corollary 5.40.

Proof.

Proposition 5.41.

Proof.

Example 5.42.

Corollary 5.43.

Proof.

Example 5.44.

Question 5.45.

Remark 5.46.

Corollary 5.47.

Proof.

Lemma 5.48.

Proof.

Theorem 5.49.

Proof.

Proposition 5.51.

Proof.

Proposition 5.52.

Proof.

Proposition 5.53.

Proof.

5.5. The one-dimensional case

Definition 5.54.

Proposition 5.55.

Proof.

Lemma 5.56.

Proof.

Lemma 5.57.

Proof.

Lemma 5.58.

Proof.

5.1. The variety with corners $X_{\Sigma}(\mathbb{R}_{\geq 0})$