The Banks of the Cohomology River
David Eisenbud, Frank-Olaf Schreyer

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.
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