Semipositive line bundles on punctured Riemann surfaces: Bergman kernels and random zeros

TL;DR

This paper investigates the properties of Bergman kernels and the distribution of zeros of Gaussian holomorphic sections for high tensor powers of semipositive line bundles on punctured Riemann surfaces, revealing key probabilistic and geometric behaviors.

Contribution

It provides new results on zero distribution, large deviations, and statistical properties of Gaussian sections in the semi-classical limit for punctured Riemann surfaces.

Findings

Zeros are equidistributed in the semi-classical limit

Large deviation estimates for zero distributions

Central limit theorem and variance results for zeros

Abstract

We give an extensive study on the Bergman kernel expansions and the random zeros associated with the high tensor powers of a semipositive line bundle on a complete punctured Riemann surface. We prove several results for the zeros of Gaussian holomorphic sections in the semi-classical limit, including the equidistribution, large deviation estimates, central limit theorem, and number variances.