Abstract
For a G-variety X with an open orbit, we define its boundary ∂X as the complement of the open orbit. The action sheaf SX is the subsheaf of the tangent sheaf made of vector fields tangent to ∂X. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups Hi(X, SX) for i > 0, extending results of F. Bien and M. Brion [BB96]. We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in [Pa08b].