Dan Coman
Syracuse University, Syracuse NY, USA

Let $L$ be a holomorphic line bundle on a compact normal complex space $X$ of dimension $n$, let $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ be an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, and $\tau=(\tau_1,\ldots,\tau_\ell)$ be an $\ell$-tuple of positive real numbers.

We 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 $\Sigma_{j}$, $1\leq j\leq\ell$, and give necessary and sufficient conditions to ensure that $\dim H^0_0(X,L^p)\sim p^n$. We also discuss the convergence of the corresponding Fubini-Study currents and their potentials, and the distribution of normalized currents of integration along zero divisors of random holomorphic sections in $H^0_0 (X, L^p)$ as $p\to\infty$. This is joint work with George Marinescu and Viêt-Anh Nguyên.