Holomorphic sections of line bundles vanishing along subvarieties

TL;DR

This paper investigates the asymptotic behavior of holomorphic sections of line bundles on complex spaces that vanish along specified subvarieties, providing criteria for their growth, estimates, and distribution properties as the tensor power increases.

Contribution

It establishes necessary and sufficient conditions for the dimension growth of sections vanishing along subvarieties, extending Ji-Shiffman's criterion to this setting.

Findings

Dimension of sections grows like p^n under certain conditions.

Provides estimates for the partial Bergman kernel.

Analyzes convergence of Fubini-Study currents and zero divisor distributions.

Abstract

Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a holomorphic line bundle on $X$. Suppose $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_\ell)$ is an $\ell$-tuple of positive real numbers, and consider the space $H^0_0 (X, L^p)$ of global holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $\tau_{j}p$ along $\Sigma_{j}$, $1\leq j\leq\ell$. We find necessary and sufficient conditions which ensure that $\dim H^0_0(X,L^p)\sim p^n$, analogous to Ji-Shiffman's criterion for big line bundles. We give estimates of the partial Bergman kernel, investigate the convergence of the Fubini-Study currents and their potentials, and the equilibrium distribution of normalized currents of integration along zero divisors of random holomorphic sections in $H^0_0 (X, L^p)$ as $p\to\infty$. Regularity results for the equilibrium envelope are also included.