Off-diagonal decay of toric Bergman kernels

TL;DR

This paper investigates the off-diagonal decay behavior of Bergman and Berezin kernels on compact toric Kähler manifolds, establishing asymptotic formulas under different regularity conditions of the metric.

Contribution

It extends understanding of kernel decay from real analytic to smooth metrics, providing new asymptotic estimates and bounds for the Berezin kernels.

Findings

Asymptotic formula for Berezin kernels with real analytic metrics.

Decay bounds for Berezin kernels with smooth metrics.

Comparison with previous negative results by Mike Christ.

Abstract

We study the off-diagonal decay of Bergman kernels $\Pi_{h^k}(z,w)$ and Berezin kernels $P_{h^k}(z,w)$ for ample invariant line bundles over compact toric projective \kahler manifolds of dimension $m$. When the metric is real analytic, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ where $D(z,w)$ is the diastasis. When the metric is only $C^{\infty}$ this asymptotic cannot hold for all $(z,w)$ since the diastasis is not even defined for all $(z,w)$ close to the diagonal. We prove that for general $C^{\infty}$ metrics, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ as long as $w$ lies on the ${\mathbb R}_+^m$-orbit of $z$, and for general $(z,w)$, $\limsup_{k \to \infty} \frac{1}{k} \log P_{h^k}(z,w) \leq - D(z^*,w^*)$ where $D(z, w^*)$ is the diastasis between $z$ and the translate of $w$ by $(S^1)^m$ to the ${\mathbb R}_+^m$ orbit of $z$, complementary to Mike Christ's negative results (arXiv:1308.5644).