Local rigidity of quasi-regular varieties

TL;DR

This paper investigates the local rigidity of certain smooth spherical varieties with Picard number one by analyzing the vanishing of specific cohomology groups related to boundary tangent sheaves.

Contribution

It extends previous results by proving cohomology vanishing for a large family of smooth spherical varieties, aiding in understanding their local rigidity.

Findings

Vanishing of higher cohomology groups for boundary tangent sheaves in many smooth spherical varieties

Extension of prior results by Bien and Brion to broader class of varieties

Application to classify local rigidity of specific projective varieties

Abstract

For a $G$-variety $X$ with an open orbit, we define its boundary $\partial X$ as the complement of the open orbit. The action sheaf $S_X$ is the subsheaf of the tangent sheaf made of vector fields tangent to $\partial X$. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups $H^i(X,S_X)$ for $i>0$, extending results of F. Bien and M. Brion. We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in a previous paper of the first author.