On the first order asymptotics of partial Bergman kernels

TL;DR

This paper investigates the asymptotic behavior of partial Bergman kernels, demonstrating exponential decay near the vanishing locus and providing uniform estimates for singular metrics, with implications for complex geometry.

Contribution

It establishes general conditions for exponential decay of partial Bergman kernels and derives uniform estimates for singular metrics along hypersurfaces.

Findings

Exponential decay of partial Bergman kernels near the vanishing locus

Uniform estimates for Bergman kernels with singular metrics

Asymptotic analysis on compact sets and near the vanishing locus

Abstract

We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metric along the hypersurface. Finally, we study the asymptotics of the partial Bergman kernel function on a given compact set and near the vanishing locus.