The Banks of the Cohomology River

TL;DR

This paper establishes precise bounds on the cohomology vanishing of tensor products of vector bundles on projective space, introduces generalized regularity indices, and explores deformation theory based on cohomology tables.

Contribution

It introduces generalized regularity indices for vector bundles and provides bounds on cohomology vanishing, advancing understanding of deformation theory.

Findings

Derived sharp bounds for cohomology vanishing of tensor products

Introduced generalized Castelnuovo-Mumford regularity indices

Constructed families of bundles with specific cohomology tables

Abstract

We give sharp bounds on the vanishing of the cohomology of a tensor product of vector bundles on the n-dimensional projective space 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.