Classical sheaf cohomology rings on Grassmannians

TL;DR

This paper computes the sheaf cohomology ring structure, called polymology, for deformations of the tangent bundle over Grassmannians, advancing the understanding of quantum sheaf cohomology in algebraic geometry.

Contribution

It provides the first explicit computation of the sheaf cohomology ring for deformed tangent bundles on Grassmannians, extending classical cohomology results.

Findings

Computed the ring structure of sheaf cohomology for deformed tangent bundles.

Established foundational results for quantum sheaf cohomology of Grassmannians.

Connected mathematical results with physical conjectures and examples.

Abstract

Let the vector bundle $\mathcal{E}$ be a deformation of the tangent bundle over the Grassmannian $G(k,n)$. We compute the ring structure of sheaf cohomology valued in exterior powers of $\mathcal{E}$, also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [arXiv:1512.08586] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples.