Gaussian holomorphic sections on noncompact complex manifolds

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.