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Register a new userConstructing quotients of algebraic varieties by linear algebraic group actions
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In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumfords geometric invariant theory (GIT). The article surveys some recent work on geometric invariant theory and quotients of varieties by linear algebraic group actions, as well as background material on linear algebraic groups, Mumfords GIT and some of the challenges that the non-reductive setting presents. The earlier work of two of the authors in the setting of unipotent group actions is extended to deal with actions of any linear algebraic group. Given the data of a linearisation for an action of a linear algebraic group H on an irreducible variety $X$, an open subset of stable points $X^s$ is defined which admits a geometric quotient variety $X^s/H$. We construct projective completions of the quotient $X^s/H$ by considering a suitable extension of the group action to an action of a reductive group on a reductive envelope and using Mumfords GIT. In good cases one can also compute the stable locus $X^s$ in terms of stability (in the sense of Mumford for reductive groups) for the reductive envelope.
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An textit{algebraic} action of a discrete group $Gamma $ is a homomorphism from $Gamma $ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $Gamma $ is determined by a module $M=widehat{X}$ over the integer group ring $mathbb{Z}Gamma $ of $Gamma $. The simplest examples of such modules are of the form $M=mathbb{Z}Gamma /mathbb{Z}Gamma f$ with $fin mathbb{Z}Gamma $; the corresponding algebraic action is the textit{principal algebraic $Gamma $-action} $alpha _f$ defined by $f$. In this note we prove the following extensions of results by Hayes cite{Hayes} on ergodicity of principal algebraic actions: If $Gamma $ is a countably infinite discrete group which is not virtually cyclic, and if $finmathbb{Z}Gamma $ satisfies that right multiplication by $f$ on $ell ^2(Gamma ,mathbb{R})$ is injective, then the principal $Gamma $-action $alpha _f$ is ergodic (Theorem ref{t:ergodic2}). If $Gamma $ contains a finitely generated subgroup with a single end (e.g. a finitely generated amenable subgroup which is not virtually cyclic), or an infinite nonamenable subgroup with vanishing first $ell ^2$-Betti number (e.g., an infinite property $T$ subgroup), the injectivity condition on $f$ can be replaced by the weaker hypothesis that $f$ is not a right zero-divisor in $mathbb{Z}Gamma $ (Theorem ref{t:ergodic1}). Finally, if $Gamma $ is torsion-free, not virtually cyclic, and satisfies Linnells textit{analytic zero-divisor conjecture}, then $alpha _f$ is ergodic for every $fin mathbb{Z}Gamma $ (Remark ref{r:analytic zero divisor}).
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-li
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