Central Limit theorem for toric \kahler manifolds

TL;DR

This paper proves a central limit theorem for measures derived from Bergman kernels on polarized toric Kähler manifolds, showing they converge to Gaussian distributions after re-centering and scaling, with explicit variance and error estimates.

Contribution

It establishes a new CLT for Bergman kernel measures on toric Kähler manifolds, including Berry-Esseen type remainder estimates, using only Kähler analysis.

Findings

Measures converge to Gaussian after re-centering and scaling.

Variance of the Gaussian is given by the Hessian of the Kähler potential.

Provides explicit error bounds of Berry-Esseen type.

Abstract

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, \omega, L, h)$ are sequences of measures $\{\mu_k^z\}_{k=1}^{\infty}$ parametrized by the points $z \in M$. For each $z$ in the open orbit, we prove a central limit theorem for $\mu_k^z$. The center of mass of $\mu_k^z$ is the image of $z$ under the moment map; after re-centering at $0$ and dilating by $\sqrt{k}$, the re-normalized measure tends to a centered Gaussian whose variance is the Hessian of the \kahler potential at $z$. We further give a remainder estimate of Berry-Esseen type. The sequence $\{\mu_k^z\}$ is generally not a sequence of convolution powers and the proofs only involve \kahler analysis.