Affine embeddings of homogeneous spaces

Details Affine embeddings of homogeneous spaces Affine embeddings of homogeneous spaces

2005-03-24

arxiv.org/abs/math/0503563v1

Surveys in geometry and number theory, 1--51, LMS Lecture Note Ser., 338, Cambridge Univ. Press, 2007

Mathematics

Algebraic Geometry

40 pages, 72 references, to appear in the LMS Lecture Notes Series

Scientific paper

Let G be a reductive algebraic group and H a closed subgroup of G. An affine embedding of the homogeneous space G/H is an affine G-variety with an open G-orbit isomorphic to G/H. We start with some basic properties of affine embeddings and consider the cases, where the theory is well-developed: toric varieties, normal SL(2)-embeddings, S-varieties, and algebraic monoids. We discuss connections between the theory of affine embeddings and Hilbert's 14th Problem via a theorem of Grosshans. We characterize affine homogeneous spaces G/H such that any affine embedding of G/H contains a finite number of G-orbits. The maximal value of modality over all affine embeddings of a given affine homogeneous space is computed and the group of equivariant automorphisms of an embedding is studied. As applications of the theory, we describe invariant algebras on homogeneous spaces of a compact Lie group and G-algebras with finitely generated invariant subalgebras.

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