Quantitative upper bounds for Bergman kernels associated to smooth K\"ahler potentials

TL;DR

This paper establishes quantitative upper bounds for Bergman kernels linked to smooth Kähler potentials, improving decay estimates for various classes of potentials including analytic and Gevrey.

Contribution

It provides new upper bounds for Bergman kernels based on the growth of Taylor coefficients of Kähler potentials, enhancing understanding of their decay properties.

Findings

Improved off-diagonal decay rates for analytic potentials

Extended bounds to quasi-analytic and Gevrey classes

Quantitative estimates based on Taylor coefficient growth

Abstract

We give upper bounds for the Bergman kernels associated to tensor powers of a smooth positive line bundle in terms of the rate of growth of the Taylor coefficients of the K\"ahler potential. As applications, we obtain improved off-diagonal rate of decay for the classes of analytic, quasi-analytic, and more generally Gevrey potentials.