Holomorphic sections of line bundles 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 $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$ 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 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 ${\Sigma }_j$, $1\le j\le \ell$. 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.

34 pages

References in corpus (2)

• Bergman kernels and equilibrium measures for ample line bundles
• Interface asymptotics of partial Bergman kernels on $S^1$-symmetric Kaehler manifolds