Subsection 3.3 (00TI) · ScalingStacks

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3.3 Kählerian polarisation

We specify a polarisation class $[\Delta]$ on the toric manifold $\mathbb{CP}^{n+1}=\mathbb{P}_{\Delta}$ . A standard background Kähler metric is (a suitable multiple of) the Fubini-Study metric:

\begin{split}\omega_{FS}&=\frac{\sqrt{-1}(n+2)}{2}\partial\bar{\partial}\log(|Z_{0}|^{2}+\ldots|Z_{n+1}|^{2})\\ &=\frac{\sqrt{-1}(n+2)}{2}\partial\bar{\partial}\log(\sum_{m\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m,\text{Log}(z)\rangle}).\end{split}

Our normalisation guarantees that the potential has the asymptotic behaviour

\sup_{z}|\frac{(n+2)}{2}\log(\sum_{m\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m,\text{Log}(z)\rangle})-\max_{m\in\Delta}\langle m,\text{Log}(z)\rangle|<+\infty.

A general (singular) Kähler metric $\omega_{u}$ on $(\mathbb{P}_{\Delta},[\Delta])$ is given by a relative potential $u\in PSH(X,\omega_{FS})$ . Alternatively, one thinks of $\omega_{u}$ as a collection of local absolute potentials:

\begin{cases}u_{0}=u+\frac{(n+2)}{2}\log(\sum_{m^{\prime}\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle}),\\ u_{m}=u+\frac{(n+2)}{2}\log(\sum_{m^{\prime}\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle})-\langle m,\text{Log}(z)\rangle,\end{cases}

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where $u_{0}$ is a local potential in a compact region, and $u_{m}$ give the local potentials near the toric boundary.

We call a convex function $u$ on $N_{\mathbb{R}}=\mathbb{R}^{n+1}$ admissible if it satisfies the asymptotic growth condition

\sup_{x}|u(x)-\max_{m\in\Delta}\langle m,x\rangle|<+\infty,

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which captures the information of the Kähler class.

00QP

Proposition 3.16. A convex function $u$ is admissible if and only if the Kähler current defined by the psh function $u\circ\text{Log}$ on $(\mathbb{C}^{*})^{n+1}$ extends to a torus invariant Kähler current on $(\mathbb{P}_{\Delta},[\Delta])$ with continuous local potentials.

00QQ

Proof. (Sketch) Convex functions on $(\mathbb{C}^{*})^{n}$ correspond to torus invariant psh functions via the log map (cf. Lemma 4.3 below). If $u$ is admissible, then near the toric boundary the appropriate local potential $u_{m}$ extends continuously over the boundary piece by the growth asymptote assumption and convexity, and the extension remains psh. Conversely, the asymptotic condition is dictated by the local boundedness of $u_{m}$ near the toric boundary pieces. ∎

A general (singular) Kähler metric $\omega_{\varphi}$ on $X_{s}$ in the polarisation class $s^{-1}[\Delta]$ is given by a potential $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ . The normalising factor $s^{-1}$ is aimed at extracting nontrivial limits as $s\to\infty$ . We can completely analogous define the local potentials:

\begin{cases}\varphi_{0}=\varphi+\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex{(\Delta)}}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle}),\\ \varphi_{m}=\varphi+\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex{(\Delta)}}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle})-\langle m,\text{Log}_{s}(z)\rangle,\end{cases}

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which are by definition psh on respective regions.

In particular, we can represent the Calabi-Yau metric $\omega_{CY,s}$ on $X_{s}$ by a potential $\varphi_{CY,s}$ . The Calabi-Yau condition is

\omega_{CY,s}^{n}=a_{s}s^{-n}d\mu_{s},

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where the normalising constant

a_{s}=\frac{\int_{X_{s}}[\Delta]^{n}}{\int_{X_{s}}d\mu_{s}}\to a_{\infty}=\frac{\int_{X_{s}}[\Delta]^{n}}{\text{Vol}(\partial\Delta_{\lambda}^{\vee})}

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as $s\to+\infty$ (cf. Prop. 3.14).