Action Of Reductive Group Research Articles

Homogeneous Kähler and Hamiltonian manifolds

We consider actions of reductive complex Lie groups $G=K^C$ on K\"ahler manifolds $X$ such that the $K$--action is Hamiltonian and prove then that the closures of the $G$--orbits are complex-analytic in $X$. This is used to characterize reductive homogeneous K\"ahler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit $K$--moment maps if and only if their isotropy groups are algebraic.

Cohomological properties of invariant quotients of affine spaces

We compute the Chow and the cohomology rings of varieties that are obtained as the GIT-quotient of affine spaces by actions of reductive groups. We prove the vanishing of the higher cohomology groups of numerically effective line bundles. As an application we prove a numerical effectiveness result for direct images of numerically effective line bundles.

Convexity properties of gradient maps

We consider the action of a real reductive group G on a Kähler manifold Z which is the restriction of a holomorphic action of a complex reductive group H. We assume that the action of a maximal compact subgroup U of H is Hamiltonian and that G is compatible with a Cartan decomposition of H. We have an associated gradient map μ p : Z → p where g = k ⊕ p is the Cartan decomposition of g . For a G-stable subset Y of Z we consider convexity properties of the intersection of μ p ( Y ) with a closed Weyl chamber in a maximal abelian subspace a of p . Our main result is a Convexity Theorem for real semi-algebraic subsets Y of Z = P ( V ) where V is a unitary representation of U.

A quotient restriction theorem for actions of real reductive groups

We prove a version of the Chevalley Restriction Theorem for the action of a real reductive group G on a topological space X which locally embeds into a holomorphic representation. Assuming that there exists an appropriate quotient X // G for the G -action, we introduce a stratification which is defined with respect to orbit types of closed orbits. Our main result is a description of the quotient X // G in terms of quotients by normalizer subgroups associated to the stratification.

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.

Nonlinearizable actions of commutative reductive groups

We generalize a construction of Freudenburg and Moser-Jauslin in order to obtain an example of a nonlinearizable action of a commutative reductive group on the affine space for every field k of characteristic zero which admits a quadratic extension.

Geometric invariant theory via Cox rings

We consider actions of reductive groups on a variety with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand–MacPherson type correspondences relating quotients by reductive groups to quotients by torus actions. Moreover, our approach provides a general access to the geometry of many of the resulting quotient spaces.

Morse and semistable stratifications of Kähler spacesby ${\Bbb C}^{\ast}$ -actions

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a \({\Bbb C}^{\ast}\)-Kahler manifold coincide if the moment map is proper and if the fixed points set \(X^{{\Bbb C}^{\ast}}\) has a finite number of connected components. For general Kahler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan.

Stratifications with respect to actions of real reductive groups

AbstractWe study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.

Two-dimensional quotients of Cn are isomorphic to C2/Γ

We will prove that any two-dimensional quotient of an affine space modulo a reductive algebraic group is isomorphic to a quotient of C2 modulo a finite group. The proof uses some new results due to Koras and Russell on contractible surfaces with at most quotient singularities and also several results about reductive group actions on affine varieties.

Hessians of the Calabi Functional and the Norm Function

We first present a new proof of a result of Calabi, nonnegativeness of the Hessian of the Calabi functional at an extremal Kahler metric. Then we extend the result to the general context of a reductive group action on a Kahler manifold admitting a moment map. Precisely, we show nonnegativeness of the Hessian of the norm function along a complex orbit at a critical point.

Symplectic slices for actions of reductive groups

Let be a reductive algebraic group over the field , a symplectic smooth affine algebraic variety, a Hamiltonian action, a point in with closed orbit. The structure of the variety in some invariant neighbourhood of the point is described. The neighbourhood is taken in the complex topology.

The index of representations associated with stabilisers

Let Q be an algebraic group with Lie algebra q and V a finite-dimensional Q-module. The index of V, denoted ind ( q , V ) , is the minimal codimension of the Q-orbits in the dual space V ∗ . By Vinberg's inequality, ind ( q , V ∗ ) ⩽ ind ( q v , ( V / q ⋅ v ) ∗ ) for any v ∈ V . In this article, we study conditions that guarantee equality. In case of reductive group actions, we show that it suffices to test the nilpotent elements in V and all its slice representations. It was recently proved by J.-Y. Charbonnel that the equality for indices holds for the adjoint representation of a semisimple group. Another proof for the classical series was given by the second author. One of our goals is to understand what is going on in the case of isotropy representations of symmetric spaces.

On motivic decompositions arising from the method of Białynicki-Birula

Recently, V. Chernousov, S. Gille and A. Merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the Bruhat decomposition. Using the method of Bialynicki-Birula, I generalize this decomposition to the case of a (possibly anisotropic) smooth projective variety homogeneous under the action of an isotropic reductive group. This answers a question of N. Karpenko.

Moduli of affine schemes with reductive group action

For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subvarieties, up to the action of the centralizer of G in GL(V). As a consequence, there are only finitely many isomorphism classes of affine G-varieties affording a prescribed multiplicity-free representation in their coordinate ring. Final version, to appear in Journal of Algebraic Geometry

On the birational classification of algebraic group actions

In this paper we show that, in a number of cases, the birational classification of some actions of an arbitrary algebraic group can be reduced to the birational classification of locally free actions of reductive groups. We prove the existence of ordinary and relative sections for some actions

A generalization of Mumford's geometric invariant theory

We generalize Mumford's construction of good quotients for reductive group actions. Replacing a single linearized invertible sheaf with a certain group of sheaves, we obtain a Geometric Invariant Theory producing not only the quasiprojective quotient spaces, but more generally all divisorial ones. As an application, we characterize in terms of the Weyl group of a maximal torus, when a proper reductive group action on a smooth complex variety admits an algebraic variety as orbit space.

The Quantization Conjecture Revisited

A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X, L), the cohomologies of L over the GIT quotient X//G equal the invariant part of the cohomologies over X. This generalizes the theorem of [GS] on global sections, and strengthens its subsequent extensions ([JK], [li]) to RiemannRoch numbers. Remarkable by-products are the invariance of cohomology of vector bundles over X//G under a small change in the defining polarization or under shift desingularization, as well as a new proof of Boutot's theorem. Also studied are equivariant holomorphic forms and the equivariant Hodgeto-de Rham spectral sequences for X and its strata, whose collapse is shown. One application is a new proof of the Borel-Weil-Bott theorem of [Ti] for the moduli stack of C-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Collapse of the Hodge-to-de Rham sequences for these stacks is also shown.

On fixed points of algebraic actions on ℂ n

It is shown that the action regarded for a rather long time by experts as a possible example disproving the conjecture on the existence of fixed points for reductive algebraic group actions on affine spaces is not an action on an affine variety, and therefore provides no example of this kind. Moreover, it is shown that the actions naturally related to the original one provide no examples of this kind as well.

A convexity theorem for noncommutative gradient flows

In this survey we shall prove a convexity theorem for gradient actions of reductive Lie groups on Riemannian symmetric spaces. After studying general properties of gradient maps, this proof is established by (1) an explicit calculation on the hyperbolic plane followed by a transfer of the results to general reductive Lie groups, (2) a reduction to a problem on abelian spaces using Kostant's Convexity Theorem, (3) an application of Fenchel's Convexity Theorem. In the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.