5.7. Approachable and integrable metrics [02VK]

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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.