Reductive group actions on affine varieties and their doubling

Dmitri I. Panyushev

Displaying similar documents to “Reductive group actions on affine varieties and their doubling”

On deformation method in invariant theory

Dmitri Panyushev (1997)

Annales de l'institut Fourier

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In this paper we relate the deformation method in invariant theory to spherical subgroups. Let $G$ be a reductive group, $Z$ an affine $G$ -variety and $H \subset G$ a spherical subgroup. We show that whenever $G / H$ is affine and its semigroup of weights is saturated, the algebra of $H$ -invariant regular functions on $Z$ has a $G$ -invariant filtration such that the associated graded algebra is the algebra of regular functions of some explicit horospherical subgroup of $G$ . The deformation method in its usual form,...

Fixed points for reductive group actions on acyclic varieties

Martin Fankhauser (1995)

Annales de l'institut Fourier

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Let $X$ be a smooth, affine complex variety, which, considered as a complex manifold, has the singular $ℤ$ -cohomology of a point. Suppose that $G$ is a complex algebraic group acting algebraically on $X$ . Our main results are the following: if $G$ is semi-simple, then the generic fiber of the quotient map $π : X \to X / / G$ contains a dense orbit. If $G$ is connected and reductive, then the action has fixed points if $\dim X / / G \leq 3$ .

Introduction to actions of algebraic groups

Michel Brion (2010)

Les cours du CIRM

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These notes present some fundamental results and examples in the theory of algebraic group actions, with special attention to the topics of geometric invariant theory and of spherical varieties. Their goal is to provide a self-contained introduction to more advanced lectures.

Linear models for reductive group actions on affine quadrics

Michael Doebeli (1994)

Bulletin de la Société Mathématique de France

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Stable affine models for algebraic group actions.

Reichstein, Zinovy, Vonessen, Nikolaus (2004)

Journal of Lie Theory

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Uniqueness properties for spherical varieties

Ivan Losev (2010)

Les cours du CIRM

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The goal of these lectures is to explain speaker’s results on uniqueness properties of spherical varieties. By a uniqueness property we mean the following. Consider some special class of spherical varieties. Define some combinatorial invariants for spherical varieties from this class. The problem is to determine whether this set of invariants specifies a spherical variety in this class uniquely (up to an isomorphism). We are interested in three classes: smooth affine varieties, general...