Borel-Weil factorization for super Grassmannians

TL;DR

This paper computes the cohomology of Schur functors on super Grassmannians, revealing a free module structure and explicit generators as irreducible representations of the supergroup, using commutative algebra techniques.

Contribution

It introduces a novel method to compute cohomology on super Grassmannians by relating it to Tor groups, providing explicit descriptions of generators as supergroup representations.

Findings

Cohomology is a free module over the structure sheaf's cohomology in certain cases

Generators form an irreducible representation of the general linear supergroup

Techniques connect cohomology calculations to algebraic Tor groups

Abstract

This article deals with computing the cohomology of Schur functors applied to tautological bundles on super Grassmannians. We show that in a range of cases, the cohomology is a free module over the cohomology of the structure sheaf and that the space of generators is an irreducible representation of the general linear supergroup that can be constructed via explicit multilinear operations. Our techniques come from commutative algebra: we relate this cohomology calculation to Tor groups of certain algebraic varieties.