Upper bounds for Bergman kernels associated to positive line bundles with smooth Hermitian metrics

TL;DR

This paper derives optimal off-diagonal upper bounds for Bergman kernels associated with high tensor powers of positive holomorphic line bundles over compact complex manifolds, assuming smooth Hermitian metrics with positive curvature.

Contribution

It provides the first sharp off-diagonal bounds for Bergman kernels with smooth, non-analytic metrics, extending previous results to a broader class of metrics.

Findings

Established optimal off-diagonal bounds for Bergman kernels

Bounds are valid for smooth Hermitian metrics with positive curvature

Results are asymptotic as tensor power tends to infinity

Abstract

Off-diagonal upper bounds are established away from the diagonal for the Bergman kernels associated to high powers of holomorphic line bundles over compact complex manifolds, asymptotically as the power tends to infinity. The line bundle is assumed to be equipped with a Hermitian metric with positive curvature form, which is infinitely differentiable but not necessarily real analytic. The bounds obtained are the best possible for this class of metrics.