Hochschild cohomology of generalised Grassmannians

TL;DR

This paper computes the Hochschild cohomology of generalized Grassmannians, revealing concentration in global sections for certain types and providing explicit descriptions of algebraic structures, with implications for algebraic geometry.

Contribution

It provides the first explicit Hochschild-Kostant-Rosenberg decomposition for generalized Grassmannians, especially for (co)minuscule and (co)adjoint cases, and describes their Gerstenhaber algebra structure.

Findings

Hochschild cohomology decomposes in specific cases

Higher cohomology vanishes for certain Grassmannians

Explicit algebraic structures are described for key cases

Abstract

We compute the Hochschild-Kostant-Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e. partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group. We explain how the decomposition is concentrated in global sections for so-called (co)minuscule and (co)adjoint generalised Grassmannians, and we suggest that for (almost) all other cases the same vanishing of the higher cohomology does not hold. Our methods give an explicit partial description of the Gerstenhaber algebra structure for the Hochschild cohomology of cominuscule and adjoint generalised Grassmannians. We also consider the case of adjoint partial flag varieties in type A, which are associated to certain submaximal parabolic subgroups.