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


Steve Zelditch (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)

Pierre Flurin (École Normal Superieure, Paris, France)


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.


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