Local rigidity of quasi-regular varieties

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.