Convergence of Fubini-Study currents for orbifold line bundles

TL;DR

This paper proves that Fubini-Study currents for high powers of semipositive singular Hermitian orbifold line bundles converge to the curvature current, extending Tian's theorem, with applications to zeros of random holomorphic sections.

Contribution

It generalizes Tian's convergence theorem to orbifold line bundles with singular metrics, providing new insights into their asymptotic behavior.

Findings

Fubini-Study currents converge weakly to the curvature current where positive

Extension of Tian's theorem to orbifold line bundles

Applications to zero distribution of random holomorphic sections

Abstract

We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.