The banks of the cohomology river

Abstract

We give sharp bounds on the vanishing of the cohomology of a tensor product of vector bundles on ${\mathbb{P}}^n$ in terms of the vanishing of the cohomology of the factors. For this purpose we introduce regularity indices generalizing the Castelnuovo–Mumford regularity.

As an application we give a sufficient condition for a vector bundle to have an unobstructed deformation theory that depends only on the cohomology table of the bundle. We construct complete families of bundles with such cohomology tables.