Variance Estimate of Systems of Random Holomorphic Sections in a Sequence of Line Bundles on Compact Kahler Manifolds

TL;DR

This paper provides a comprehensive variance estimate for zeros of random holomorphic sections in line bundles over compact Kahler manifolds, extending to general probability measures including Gaussian and Fubini-Study.

Contribution

It introduces a broad variance estimation framework applicable to a wide class of probability measures on holomorphic sections.

Findings

Variance estimates for zeros of random sections

Applicability to Gaussian and Fubini-Study measures

Generalization to a broad class of probability measures

Abstract

This paper primarily concerns the variance estimate of zeros of systems of random holomorphic sections associated with a sequence of smooth Hermitian holomorphic line bundles on a compact Kahler manifold X. The probability measures taken into consideration in this paper satisfy a certain condition which makes it much more general(possibly the most general) than (all) the measures mostly studied in the literature. In particular we also give variance estimates for several known measures such as Gaussian and Fubini-Study measures.