Asymptotic growth of global sections on open varieties

Abstract

Let $X$ be a projective variety and let $E$ be a reduced divisor. We study the asymptotic growth of the dimension of the space of global sections of powers of a divisor $D$ on $X\backslash E$. We show that it is always bounded by a polynomial of degree $\dim(X)$, if finite. Furthermore, when $D$ is big, we characterize the finiteness of the cohomology groups in question. This answers a question of Zariski and Koll\'ar.