Holomorphic Line Bundles over a Tower of Coverings

TL;DR

This paper investigates the properties of holomorphic line bundles over a sequence of coverings of a compact Kähler manifold, establishing stability, equidistribution, and convergence results for random holomorphic sections.

Contribution

It provides effective estimates for Bergman stability and zero current equidistribution, along with variance bounds and almost sure convergence under geometric conditions.

Findings

Effective estimate implying Bergman stability

Equidistribution of zero currents of random sections

Variance estimate leading to almost sure convergence

Abstract

We study a tower of normal coverings over a compact K\"ahler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furthermore, we obtain a variance estimate for those random zero currents, which yields the almost sure convergence under some geometric condition.