Logarithmic Bergman kernel and Conditional expectation of Gaussian holomorphic fields

TL;DR

This paper establishes the asymptotic behavior of the logarithmic Bergman kernel and applies it to compute the conditional expectation of zero densities of Gaussian holomorphic sections vanishing along a subvariety.

Contribution

It provides the first asymptotic analysis of the logarithmic Bergman kernel and links it to the conditional zero density of Gaussian holomorphic fields.

Findings

Asymptotic formula for the logarithmic Bergman kernel

Explicit calculation of the conditional zero density

Connection between Bergman kernel asymptotics and Gaussian field zeros

Abstract

We prove the asymptotic of the logarithmic Bergman kernel. And as an application, we calculate the conditional expectation of density of zeros of Gaussian random sections of powers of a positive line bundle that vanish along a fixed smooth subvariety.