H\"older singular metrics on big line bundles and equidistribution

TL;DR

This paper proves that zeros of random sections of big line bundles with H"older singular metrics on compact Kähler manifolds become evenly distributed according to the wedge product of curvature currents, with convergence speed estimates.

Contribution

It establishes asymptotic equidistribution of zeros for random sections of big line bundles with H"older singular metrics and provides convergence rate estimates.

Findings

Zeros of random sections distribute according to curvature currents.

Asymptotic equidistribution is proven for singular Hermitian metrics.

Speed of convergence is estimated for H"older singular metrics.

Abstract

We show that normalized currents of integration along the common zeros of random $m$-tuples of sections of powers of $m$ singular Hermitian big line bundles on a compact K\"ahler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are H\"older with singularities we also estimate the speed of convergence.