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 $H_{(2)}^0(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of $L$, which, if $\dim H_{(2)}^0(X,L)=\infty$, 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.

47 pages

References in corpus (5)

• Asymptotic distribution of complex zeros of random analytic functions
• Toeplitz operators on symplectic manifolds
• Universality results for zeros of random holomorphic sections
• Hölder singular metrics on big line bundles and equidistribution
• Asymptotic normality of linear statistics of zeros of random polynomials