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# Pure and Applied Mathematics Quarterly

## Volume 18 (2022)

### Number 1

### Special Issue in Honor of Bernie Shiffman

Guest Editors: Yuan Yuan, Christopher Sogge, and Steven Morris Zelditch

### Entropy of Bergman measures of a toric Kaehler manifold

Pages: 269 – 303

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n1.a8

#### 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 $z \in M$. We determine the asymptotics of the entropies $H(\mu^z_k)$ of these measures. The sequence $\mu^z_k$ in some ways resembles a sequence of convolution powers; we determine precisely when it actually is such a sequence. When $(M,\omega)$ is a Fano toric manifold with positive Ricci curvature, we show that there exists a unique point $z_0$ (up to the real torus action) for which $\mu^z_k$ has asymptotically maximal entropy. If the Kähler metric is Kähler–Einstein, we show that the image of $z_0$ under the moment map is the center of mass of the polytope. We also show that the Gaussian measure on the space $H^0 (M, L^k)$ induced by the Kähler metric has maximal entropy at the balanced metric.

#### Keywords

Bergman kernel, holomorphic line bundle, measures on moment polytope

#### 2010 Mathematics Subject Classification

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

The first author’s research was partially supported by NSF grant DMS-1810747.

Received 14 January 2020

Accepted 6 August 2020

Published 10 February 2022