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International Mathematics Research Papers (2006) Vol. 2006 : article ID 46293, 57 pages, doi:10.1155/IMRP/2006/46293
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Stable spherical varieties and their moduli

Valery Alexeev and Michel Brion

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 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.



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This Article
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