Borel-Weil factorization for super Grassmannians

TL;DR

This work develops a Borel–Weil type factorization for the cohomology of Schur functors on tautological bundles over the super Grassmannian $Gr(p|q,V)$. By combining a degeneration framework based on the $\mathcal{J}$-adic filtration with Koszul and factorization-ring techniques, the authors prove that $H^ullet(X;\mathbf{S}_\alpha\mathcal{Q}\otimes\mathbf{S}_\beta(\mathcal{R}^*))$ is a free module over $H^ullet(X;\mathcal{O}_X)$ and that $H^0(X;\mathbf{S}_\alpha\mathcal{Q}\otimes\mathbf{S}_\beta(\mathcal{R}^*))$ is an irreducible $GL(m|n)$-representation $S_{[\alpha;\beta]}(V)$, constructed via a rational Schur functor, under explicit dimension constraints. The method links cohomology to Tor groups of determinantal-type varieties, yielding explicit character formulas via composite supersymmetric Schur polynomials and establishing $\,\mathcal{J}$-formality in the relevant cases. The results extend to relative settings and to super partial flag varieties, providing a robust framework for computing cohomology of homogeneous bundles on superspaces and highlighting connections between invariant theory, commutative algebra, and super-geometry.

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.

Theorems & Definitions (59)

• Theorem : preliminary version
• Remark 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Proposition 3.1
• proof
• Corollary 3.2
• Remark 3.3