A vanishing theorem for the canonical blow-ups of Grassmann manifolds

TL;DR

This paper proves a vanishing theorem for higher cohomology groups of the tangent bundle of certain blow-ups of Grassmann manifolds, demonstrating their local rigidity and advancing understanding of their geometric properties.

Contribution

It establishes a vanishing theorem for the tangent bundle cohomology of canonical blow-ups of Grassmannians, revealing their local rigidity.

Findings

Higher cohomology groups of the tangent bundle vanish

The blow-ups are locally rigid in the sense of Kodaira-Spencer

Provides new insights into the geometry of Grassmannian blow-ups

Abstract

Let $\mathcal T_{s,p,n}$ be the canonical blow-up of the Grassmann manifold $G(p,n)$ constructed by blowing up the Pl\"ucker coordinate subspaces associated with the parameter $s$. We prove that the higher cohomology groups of the tangent bundle of $\mathcal T_{s,p,n}$ vanish. As an application, $\mathcal T_{s,p,n}$ is locally rigid in the sense of Kodaira-Spencer.