Restricted spaces of holomorphic sections vanishing along subvarieties

TL;DR

This paper studies the asymptotic behavior of holomorphic sections vanishing along subvarieties in complex spaces, establishing conditions for dimension growth and convergence of associated currents, with applications to zero distribution of random sections.

Contribution

It provides new criteria for the dimension growth of sections restricted to subvarieties and analyzes the convergence of Fubini-Study currents to equilibrium currents in this setting.

Findings

Dimension of restricted sections grows like p^m under certain conditions.

Fubini-Study currents converge to equilibrium currents on subvarieties.

Zeros of random sections become equidistributed as p increases.

Abstract

Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ 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 let $H^0_0(X,L^p)$ be the space of holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $\tau_jp$ along $\Sigma_j$, $1\leq j\leq\ell$. If $Y\subset X$ is an irreducible analytic subset of dimension $m$, we consider the space $H^0_0 (X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to global holomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet $(L,\Sigma,\tau)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we give a general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When $L$ is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certain equilibrium current on $Y$. We apply this to the study of the equidistribution of zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as $p\to\infty$.