Entropy of Bergman measures of a toric Kaehler manifold

TL;DR

This paper investigates the asymptotic behavior of the entropy of measures derived from Bergman kernels on polarized toric Kähler manifolds, revealing their convolution power structure.

Contribution

It provides a detailed analysis of the entropy asymptotics of Bergman measures and characterizes when these measures form convolution powers.

Findings

Asymptotic formulas for the entropy of Bergman measures.

Conditions under which the measures are convolution powers.

Insights into the structure of Bergman measures on toric Kähler manifolds.

Abstract

Associated to the Bergman kernels of a polarized toric Kaehler manifold $(M, \omega, L, h)$ are sequences of measures $\{\mu_k^z\}_{k=1}^{\infty}$ parametrized by the points $z \in M$. We determine the asymptotics of the entropies $H(\mu_k^z)$ of these measures. The sequence $\mu_k^z$ in some ways resembles a sequence of convolution powers; we determine precisely when it actually is such a sequence.