A central limit theorem associated with a sequence of positive line bundles

Abstract

We prove a central limit theorem for smooth linear statistics associated with zero divisors of standard Gaussian holomorphic sections in a sequence of holomorphic line bundles with Hermitian metrics of class ${\mathcal{C}}^3$ over a compact Kähler manifold. In the course of our analysis, we derive first-order asymptotics and upper decay estimates for near and off-diagonal Bergman kernels, respectively. These results are essential for determining the statistical properties of the zeros of random holomorphic sections.

arXiv commentMinor revisions, 24 pages