On Orbit Closures of Borel Subgroups in Spherical Varieties

. Let F be the AEag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with ønitely many orbits, and let V be an H-orbit closure in F . Expanding the cohomology class of V in the basis of Schubert classes deønes a union V 0 of Schubert varieties in F with positive multiplicities. If G is simply-laced, we show that these multiplicites are equal to the same power of 2. For arbitrary G, we show that V 0 is connected in codimension 1. If moreover all multiplicities are 1, we show that V is normal and we construct a AEat degeneration of V to V 0 . Thus, for any eoeective line bundle L on G=B, the restriction map H 0 (G=B; L) ! H 0 (V; L) is surjective, and H i (V; L) = 0 for i 1. Introduction Let X be a spherical variety, that is, X is a normal algebraic variety endowed with an action of a connected reductive group G such that the set of orbits of a Borel subgroup B of G is ønite. These B-orbits play an important role in the geometry and top..