Stable Spherical Varieties and Their Moduli

We introduce a notion of stable spherical variety which includes the spherical varieties under a reductive group G and their flat equivariant degenerations. Given any projective space P where G acts linearly, we construct a moduli space for stable spherical varieties over P, that is, pairs (X, f), where X is a stable spherical variety and f : X → P is a finite equivariant morphism. This space is projective, and its irreducible components are rational. It generalizes the moduli space of pairs (X, D), where X is a stable toric variety and D is an effective ample Cartier divisor on X which contains no orbit. The equivariant automorphism group of P acts on our moduli space; the spherical varieties over P and their stable limits form only finitely many orbits. A variant of this moduli space gives another view to the compactifications of quotients of thin Schubert cells constructed by Kapranov and Lafforgue.