Gaussian holomorphic sections on noncompact complex manifolds

TL;DR

This paper introduces two methods for constructing Gaussian-like random holomorphic sections on noncompact complex manifolds, analyzes their zeros, and explores their distribution and probabilistic properties in semiclassical limits.

Contribution

It presents novel constructions of Gaussian random holomorphic sections, including one that yields $ ext{L}^2$-holomorphic sections via Wiener space and Berezin-Toeplitz quantization.

Findings

Construction of Gaussian random holomorphic sections on noncompact manifolds.

Analysis of zero distribution, equidistribution, and large deviations.

Identification of conditions under which sections are $ ext{L}^2$-integrable.

Abstract

We give two constructions of Gaussian-like random holomorphic sections of a Hermitian holomorphic line bundle $(L,h_{L})$ on a Hermitian complex manifold $(X,\Theta)$. In particular, we are interested in the case where the space of $\mathcal{L}^2$-holomorphic sections $H^{0}_{(2)}(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of $L$, which, if $\dim H^{0}_{(2)}(X,L)=\infty$, are almost never $\mathcal{L}^2$-integrable on $X$. The second construction combines the abstract Wiener space theory with the Berezin-Toeplitz quantization and yields a random $\mathcal{L}^2$-holomorphic section. Furthermore, we study their random zeros in the context of semiclassical limits, including their equidistribution, large deviation estimates and hole probabilities.