Cohomology of flag supervarieties and resolutions of determinantal ideals

TL;DR

This paper explores the cohomology of flag supervarieties, linking it to resolutions of determinantal ideals, and provides explicit computations for super Grassmannians, connecting supergeometry with classical algebraic geometry.

Contribution

It introduces a novel connection between the cohomology of flag supervarieties and determinantal ideal resolutions, with explicit results for super Grassmannians.

Findings

Cohomology of super Grassmannians combines Grassmannian cohomology and determinantal syzygies.

Provides a geometric framework via a super analog of the Grothendieck-Springer resolution.

Explains the action of the supergroup on determinantal ideal syzygies.

Abstract

We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we completely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated to the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail.