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# Pure and Applied Mathematics Quarterly

## Volume 17 (2021)

### Number 3

### Special Issue in Honor of Duong H. Phong

Edited by Tristan Collins, Valentino Tosatti, and Ben Weinkove

### Central limit theorem for toric Kähler manifolds

Pages: 843 – 864

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n3.a1

#### Authors

#### Abstract

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

#### Keywords

Bergman kernel, holomorphic line bundle, measures on moment polytope

#### 2010 Mathematics Subject Classification

Primary 32A25, 32L10, 60F05. Secondary 14M25, 53D20.

The research of Steve Zelditch was partially supported by NSF grants DMS-1541126 and DMS-1810747.

Received 8 January 2019

Accepted 10 July 2019

Published 14 June 2021