Towards non-reductive geometric invariant theory

@article{Doran2007TowardsNG,
  title={Towards non-reductive geometric invariant theory},
  author={Brent Doran and Frances Kirwan},
  journal={arXiv: Algebraic Geometry},
  year={2007}
}
We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the… Expand

Figures from this paper

Constructing quotients of algebraic varieties by linear algebraic group actions
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, inExpand
Projective linear configurations via non-reductive actions
We study the iterated blow-up X of projective space along an arbitrary collection of linear subspaces. By replacing the universal torsor with an $\mathbb{A}^1$-homotopy equivalent model, built fromExpand
Non-reductive geometric invariant theory and compactifications of enveloped quotients
In this thesis we develop a framework for constructing quotients of varieties by actions of linear algebraic groups which is similar in spirit to that of Mumford's geometric invariant theory. This isExpand
A pr 2 00 8 Quotients by non-reductive algebraic group actions
Geometric invariant theory (GIT) was developed in the 1960s by Mumford in order to construct quotients of reductive group actions on algebraic varieties and hence to construct and study a number ofExpand
On unipotent quotients and some A^1-contractible smooth schemes
We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principalExpand
Graded linearisations
When the action of a reductive group on a projective variety has a suitable linearisation, Mumford’s geometric invariant theory (GIT) can be used to construct and study an associated quotientExpand
3 1 M ar 2 00 8 Quotients by non-reductive algebraic group actions
Geometric invariant theory (GIT) was developed in the 1960s by Mumford in order to construct quotients of reductive group actions on algebraic varieties and hence to construct and study a number ofExpand
Quotients by non-reductive algebraic group actions
Given a suitable action on a complex projective variety X of a non-reductive affine algebraic group H, this paper considers how to choose a reductive group G containing H and a projective completionExpand
A G ] 3 0 Ja n 20 08 Quotients by non-reductive algebraic group actions
Geometric invariant theory (GIT) was developed in the 1960s by Mumford in order to construct quotients of reductive group actions on algebraic varieties and hence to construct and study a number ofExpand
Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients
We establish a method for calculating the Poincaré series of moduli spaces constructed as quotients of smooth varieties by suitable non-reductive group actions; examples of such moduli spaces includeExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 84 REFERENCES
Rational intersection cohomology of quotient varieties. II
Given a linear action of a complex reductive group G on a complex projective variety X one can define a projective "quotient" variety X//G using Mumford's geometric invariant theory [18]. From theExpand
Partial desingularisations of quotients of nonsingular varieties and their Betti numbers
When a reductive group G acts linearly on a nonsingular complex projective variety X one can define a projective "quotient" variety X//G using Mumford's geometric invariant theory. If the conditionExpand
Intersection cohomology of quotients of nonsingular varieties
Let M ⊂ P be a nonsingular projective variety acted on by a connected complex reductive group G = KC via a homomorphism G → GL(n + 1) which is the complexification of a homomorphism K → U(n+1).Expand
Variation of geometric invariant theory quotients
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes ofExpand
Equivariant intersection theory
In this paper we develop an equivariant intersection theory for actions of algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They areExpand
Geometric Quotients of Unipotent Group Actions
Let G be a unipotent algebraic group over K (a field of characteristic 0) which acts rationally on an affine scheme X = Spec A over K, where A is a commutative K-algebra. The problem of findingExpand
On quotient varieties and the affine embedding of certain homogeneous spaces
We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, r), where V/G is a variety and r: V-* V/G is a rational map, everywhere definedExpand
QUOTIENTS BY REDUCTIVE GROUP, BOREL SUBGROUP, UNIPOTENT GROUP AND MAXIMAL TORUS
Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of theExpand
Refinements of the Morse stratification of the normsquare of the moment map
Let X be any nonsingular complex projective variety on which a complex reductive group G acts linearly, and let X SS and X S be the sets of semistable and stable points of X in the sense of Mumford’sExpand
Localization for nonabelian group actions
Suppose X is a compact symplectic manifold acted on by a compact Lie group K (which may be nonabelian) in a Hamiltonian fashion, with moment map µ : X → Lie(K) ∗ and Marsden-Weinstein reduction MX =Expand
...
1
2
3
4
5
...