Asymptotic cohomological functions on projective varieties.

Abstract

In the present thesis we consider the asymptotic behavior of the cohomology groups of divisors on projective varieties. Based on the concept of the volume of a divisor, we construct a sequence of asymptotic invariants, called asymptotic cohomological functions, which are defined on the Neron-Severi space. These invariants measure the asymptotic growth of the cohomology of divisors on the variety. After having worked out the basic properties of asymptotic cohomological functions, we move on to work them out in several important special cases, notably on abelian varieties, generalized flag varieties, and smooth surfaces. We establish that in each of these cases, asymptotic cohomological functions are locally polynomial inside the cone of big divisors. We also give evidence that this no longer holds in general for higher-dimensional varieties. The main result of the thesis is the proof that asymptotic cohomological functions are continuous in general.