Bergman kernel functions associated to measures supported on totally real submanifolds

TL;DR

This paper studies the growth of Bergman kernel functions on totally real submanifolds in complex space, providing sharp bounds and applications to zero distribution of random polynomials.

Contribution

It establishes polynomial growth bounds for Bergman kernels on piecewise-smooth totally real submanifolds and extends zero distribution results to higher dimensions.

Findings

Bergman kernel functions grow polynomially on totally real submanifolds.

Bounds are sharp for smooth submanifolds.

Application to zeros of random polynomials in higher dimensions.

Abstract

We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold K in C^n is of polynomial growth (e.g, in dimension one, K is a finite union of transverse Jordan arcs in C). Our bounds are sharp when K is smooth. We give an application to equidistribution of zeros of random polynomials extending a result of Shiffman-Zelditch to the higher dimensional setting.