Central limit theorem for toric Kähler manifolds

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.

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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