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Stable spherical varieties and their moduli
Department of Mathematics, University of Georgia Athens, GA 30602, USA E-mail address: valery{at}math.uga.edu
Institut Fourier BP 74, 38402 Saint-Martin d'Hères Cedex, France E-mail address: michel.brion{at}ujf-grenoble.fr
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 
where G acts linearly, we construct a moduli space for stable spherical varieties over , that is, pairs (X,f), where X is a stable spherical variety and f:X 
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 acts on our moduli space; the spherical varieties over 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.