Universality results for zeros of random holomorphic sections

TL;DR

This paper establishes that the asymptotic distribution of zeros of random holomorphic sections is universal across different probability measures on complex spaces, with additional decay estimates for Bergman kernels on Kähler manifolds.

Contribution

It proves a universality result for zeros of random holomorphic sections on singular Hermitian line bundles and provides off-diagonal decay estimates for Bergman kernels on Kähler manifolds.

Findings

Zeros of random holomorphic sections have a universal asymptotic distribution.

The distribution is independent of the probability measure under mild conditions.

Exponential decay of Bergman kernels off-diagonal on Kähler manifolds.

Abstract

In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact K\"ahler complex space $X$. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections. In the case when $X$ is a compact K\"ahler manifold, we also prove an off-diagonal exponential decay estimate for the Bergman kernels of a sequence of positive line bundles on $X$.