Asymptotic growth of global sections on open varieties

TL;DR

This paper investigates the asymptotic behavior of global sections on open varieties, establishing polynomial bounds and characterizing finiteness conditions for big divisors, thus addressing longstanding questions in algebraic geometry.

Contribution

It provides a polynomial bound for the growth of global sections on open varieties and characterizes when certain cohomology groups are finite for big divisors.

Findings

Global sections grow at most polynomially with degree equal to the dimension of the variety.

Finiteness of cohomology groups is characterized for big divisors.

Answers to questions posed by Zariski and Kollár are provided.

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.