3.4. Kähler metrics on the toric variety [03F7]

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3.4. Kähler metrics on the toric variety

First, we would like to extend the bi-PIKAS potential to $\mathbb{R}^{d}$ by taking the Legendre transform of $\hat{\Phi}$ . Namely,

\Phi(x)=\max_{y\in{\Delta_{\nu}}}\{\langle x,y\rangle-\hat{\Phi}(y)\}.

Similarly, we extend $\hat{\Phi}$ to a function on $(\mathbb{R}^{d})^{*}$ . We will abuse the notation $\Phi,\hat{\Phi}$ for the extended potentials.

$\Phi$ is a $C^{1}$ -function, smooth when restricted to any strata of $\Delta^{\vee}_{\lambda}$ . Its Hessian $\operatorname{Hess}\Phi$ is continuous at $\partial\mathcal{U}$ , but blows off at $\partial\mathcal{V}$ . And something drastic happens at the discriminant $D=\partial\mathcal{U}\cap\partial\mathcal{V}$ .

Next we regularize the $C^{1}$ -potential $\Phi$ to get a smooth convex function on $\mathbb{R}^{d}$ which we will use later on to define a Kähler potential on the toric variety $X_{T}$ . Let $\rho$ be a mollifier with support in $-\Delta^{\vee}$ . We define $\Phi^{\mathrm{sm}}_{h}:=\rho_{h}\ast\Phi.$

Remark.

The constructed vector field, foliation, potential, etc., depend on the pair $(\lambda,\nu)$ , as well as on the regularization parameters $h,h^{\vee}$ . But to simplify the notations for $\mathfrak{X},\mathcal{F},\Phi,\Phi^{\mathrm{sm}}$ we will often leave only those indices which are important in a current consideration and omit the rest when there is no confusion possible.

Before constructing a Kähler potential on the toric variety $X_{T}$ we need another technical statement.

Lemma 3.3.

For $\tau\in T$ , the $\tau$ -slope of $\Phi^{\mathrm{sm}}_{h}$ is equal to the $\tau$ -slope of $-\nu|_{\tau}$ in some translation $R_{\geq\tau}$ of $\cup_{\tau^{\prime}\geq\tau}\operatorname{cone}(\tau^{\prime})$ .

Proof.

Consider $h=0$ first. For a simplex $\tau\in T$ , let $F_{\tau}$ be the corresponding face of ${\Delta_{\nu}}$ , and we set $R_{\tau}:=\phi\hat{\phi}^{-1}(F_{\tau})+\operatorname{cone}(\tau)$ . Then the set $\cup_{\tau^{\prime}\geq\tau}R_{\tau^{\prime}}$ contains some translation $R_{\geq\tau}$ of $\cup_{\tau^{\prime}\geq\tau}\operatorname{cone}(\tau^{\prime})$ .

On the other hand, $\Phi$ is the Legendre transform of $\hat{\Phi}|_{\Delta_{\nu}}$ . Hence, if $\nabla\Phi(x)=y$ and $r$ is in the normal cone to ${\Delta_{\nu}}$ at $y\in{\Delta_{\nu}}$ , then $\nabla\Phi(x+r)=\nabla\Phi(x)=y$ . So we see that for $x\in R_{\tau}$ the gradient $\nabla\Phi(x)$ takes values in the face $F_{\tau}$ of ${\Delta_{\nu}}$ because $x$ . In particular, the $\tau$ -slopes of $\Phi$ are equal to $-\nu|_{\tau}$ .

For $h>0$ the statement of the lemma follows from the case $h=0$ and the Proposition 3.1. The translated cones $R_{\geq\tau}$ become shifted into their interiors by some vectors of size $h$ . ∎

Now we can use $\Phi^{\mathrm{sm}}=\Phi^{\mathrm{sm}}_{h}$ with any $h>0$ to define a Kähler potential on $X_{T}$ . For an element $z=\{z_{1},\dots,z_{d}\}\in\mathbb{C}\backslash\{0\}^{d}$ we will use the notations

\log|z|:=\{\log|z_{1}|,\dots,\log|z_{d}|\}\in\mathbb{R}^{d},\quad\operatorname{Arg}(z):=\frac{1}{2\pi}\{\arg(z_{1}),\dots,\arg(z_{d})\}\in\mathbb{T}.

Proposition 3.4.

The $(1,1)$ -form defined on $(\mathbb{C}\backslash\{0\})^{d}$ by

\eta:=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\Phi^{\mathrm{sm}}\left(\log|z|\right)

extends to a smooth (in the orbifold sense) non-negative definite $(1,1)$ -form on $X_{T}$ in the cohomology class $[\eta]=[\nu]$ .

Proof.

First, we rewrite the form $\eta$ on $(\mathbb{C}\backslash\{0\})^{d}$ as

\eta=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\Phi^{\mathrm{sm}}\left(\log|z|\right)=\left\langle d\left(\nabla\Phi^{\mathrm{sm}}\left(\log|z|\right)\right)\wedge d\operatorname{Arg}(z)\right\rangle,

where ” $\langle\ \wedge\ \rangle$ ” means also the $\langle\ ,\ \rangle$ -pairing between the $(\mathbb{R}^{d})^{*}$ -valued gradient $\nabla\Phi^{\mathrm{sm}}$ and the $\mathbb{R}^{d}$ -valued 1-form $d\operatorname{Arg}(z)$ .

For a simplex $\tau\in T$ we want to show that $\eta$ extends to the toric subvariety $Z_{\tau}$ associated to $\tau$ . If $X_{T}$ is smooth, then in a neighborhood of $Z_{\tau}$ we can choose the coordinates similar to those from [HZ02, Lemma 3.9]. That is, we choose a basis $\{e_{i}\}$ such that

\langle e_{i},w_{j}\rangle=-\delta_{ij},i=1,\dots,\dim\tau+1\text{ and }\langle e_{i},\tau\rangle=0,i=\dim\tau+2,\dots,d.

Then, in the coordinates $y_{i}=z^{e_{i}}$ the equations for the subvariety $Z_{\tau}\subset X_{T}$ are $y_{i}=0,i=1,\dots,\dim\tau+1$ .

According to the theory of toric varieties (cf., e.g. [Ful93]) a neighborhood of the toric subvariety $Z_{\tau}$ lies in the closure of $\log^{-1}(R_{\geq\tau})$ , where $R_{\geq\tau}$ is any translation of the cone $\cup_{\tau^{\prime}\geq\tau}\operatorname{cone}(\tau^{\prime})$ . But by the Lemma 3.3 the directional derivatives $\langle\nabla\Phi^{\mathrm{sm}},w_{i}\rangle$ , $w_{i}\in\tau$ , are constant in some translation $R_{\geq\tau}$ of $\cup_{\tau^{\prime}\geq\tau}\operatorname{cone}(\tau^{\prime})$ . Hence, in a neighborhood of $Z_{\tau}$ the form $\eta_{a}$ written in the above coordinates is independent of $y_{i},i=1,\dots,\dim\tau+1$ , and, thus, can be extended to $Z_{\tau}$ .

In case when $X_{T}$ is an orbifold we may not be able to choose an integral basis $\{e_{i}\}$ with the above conditions. This corresponds to the fact that we may need to go to a finite cover to get a smooth form by weakening the first set of conditions to be $\langle e_{i},\tau\rangle\in\mathbb{Z}$ . But the rest of the argument goes through.

Finally, the cohomology class of a $\mathbb{T}$ -invariant $(1,1)$ -form on a complete toric variety is determined by the image of its moment map. But the moment map for $\eta$ is given on $(\mathbb{C}\backslash\{0\})^{d}$ by

\mu(z)=\nabla\Phi^{\mathrm{sm}}\left(\log|z|\right),

whose extension to the whole toric variety $X_{T}$ has the image ${\Delta_{\nu}}$ . Hence the class of $\eta$ is $[\nu]$ . ∎

Finally we can add to $\eta$ a (small) positive multiple of a Kähler (e.g., Fubini-Study) form $\omega_{0}$ . Thus we get a true Kähler form $\omega=\eta+\epsilon\omega_{0}$ on $X_{T}$ in the class $[\nu]+\epsilon[\omega_{0}])$ .