Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

TL;DR

This paper proves the convergence of measures and kernels related to weakly holomorphic sections of line bundles on algebraic curves, and studies the asymptotic zero distribution of random sections.

Contribution

It establishes the convergence of Fubini-Study measures and Bergman kernels for weakly holomorphic sections on algebraic curves, advancing understanding of their asymptotic behavior.

Findings

Normalized measures and kernels converge as sections grow large

Zeros of random sections become equidistributed in the limit

Results apply to singular Hermitian line bundles on algebraic curves

Abstract

We prove the convergence of the normalized Fubini-Study measures and the logarithms of the Bergman kernels of various Bergman spaces of holomorphic and weakly holomorphic sections associated to a singular Hermitian holomorphic line bundle on an algebraic curve. Using this, we study the asymptotic distribution of the zeros of random sequences of sections in these spaces.