Bergman kernels and equidistribution for sequences of line bundles on K\"ahler manifolds

TL;DR

This paper investigates the asymptotic behavior of Bergman kernels and the distribution of zeros for sequences of line bundles on K"ahler manifolds, providing new insights into their geometric and probabilistic properties.

Contribution

It establishes the asymptotic expansion of Bergman kernels for sequences of line bundles and analyzes the equidistribution of zeros of random sections on K"ahler manifolds.

Findings

Asymptotic expansion of Bergman kernels under curvature convergence.

Distribution of zeros of random sections becomes equidistributed as p→∞.

Results applicable to complex geometry and probabilistic analysis on manifolds.

Abstract

Given a sequence of positive Hermitian holomorphic line bundles $(L_p,h_p)$ on a K\"ahler manifold $X$, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of $L_p$, under a natural convergence assumption on the sequence of curvatures $c_1(L_p,h_p)$. We then apply this to study the asymptotic distribution of common zeros of random sequences of $m$-tuples of sections of $L_p$ as $p\to\infty$.