On the asymptotic behavior of Bergman kernels for positive line bundles

TL;DR

This paper investigates how Bergman kernels behave asymptotically for positive line bundles on complex manifolds, focusing on the influence of the metric's positivity and applications to approximating singular metrics.

Contribution

It provides new uniform estimates for Bergman kernels that depend on the positivity of the Chern form, aiding in the approximation of semi-positive singular metrics.

Findings

Established asymptotic formulas for Bergman kernels as p tends to infinity.

Analyzed the dependence of estimates on the positivity of the Chern form.

Applied results to approximate semi-positive singular metrics.

Abstract

Let L be a positive line bundle on a projective complex manifold. We study the asymptotic behavior of Bergman kernels associated with the tensor powers L^p of L as p tends to infinity. The emphasis is the dependence of the uniform estimates on the positivity of the Chern form of the metric on L. This situation appears naturally when we approximate a semi-positive singular metric by smooth positively curved metrics.