Hochschild Cohomology of Isotropic Grassmannians

TL;DR

This paper proves that certain isotropic Grassmannians are not Hochschild global, leading to the conclusion that Bott vanishing does not hold for these varieties, thus resolving a conjecture in the field.

Contribution

It establishes that nonspecial isotropic Grassmannians are not Hochschild global, confirming a conjecture and showing Bott vanishing fails for these varieties.

Findings

Nonspecial isotropic Grassmannians are not Hochschild global.

Bott vanishing fails for these Grassmannians.

Confirmed conjecture by Belmans and Smirnov.

Abstract

We prove that nonspecial isotropic Grassmannians (that is, all isotropic Grassmannians which are neither (co)adjoint nor (co)minuscule, except $\mathsf{OGr}(n-1, 2n+1)$ for $n\geq 4$), are not Hochschild global, thus establishing a conjecture by P. Belmans and M. Smirnov. As a corollary, we conclude that Bott vanishing fails for all these varieties.