Restricted spaces of holomorphic sections vanishing along subvarieties

Abstract

Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$ 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 ${\tau }_{j}p$ along ${\Sigma }_j$, $1\le j\le \ell$. 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 $p\to \infty$.

22 pages. arXiv admin note: text overlap with arXiv:1909.00328