Integration of local actions on holomorphic fiber spaces

Heinzner, Peter; Iannuzzi, Andrea

doi:10.1017/s0027763000006206

Cited by 15 publications

(24 citation statements)

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“…438] applies to show univalency of such a local action. Then by [HI,Theorem 2,p. 38] there exists a possibly non-Hausdorff universal globalization X * .…”

Section: Theorem 1 Let X Be a Taut R-manifold Then There Exists A Umentioning

confidence: 94%

“…For basic facts and results on local actions and their globalizations we refer to [P] and more generally to [HI,[1][2][3], from which most notations are inherited. However note that here all manifolds are assumed to be Hausdorff (cf.…”

Section: Existence Of Globalizationsmentioning

confidence: 99%

“…Note that every leaf Σ of Palais' foliation with respect to the induced local C-action is a non-compact Riemann surface, since its projection p| Σ : Σ → C is not constant. In particular Σ is holomorphically separable and [HI,Corollary,p. 438] applies to show univalency of such a local action.…”

Section: Theorem 1 Let X Be a Taut R-manifold Then There Exists A Umentioning

confidence: 99%

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Induced local actions on taut and Stein manifolds

Iannuzzi

2003

Proc. Amer. Math. Soc.

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Abstract. Let G = (R, +) act by biholomorphisms on a taut manifold X. We show that X can be regarded as a G-invariant domain in a complex manifold X * on which the universal complexification (C, +) of G acts. If X is also Stein, an analogous result holds for actions of a larger class of real Lie groups containing, e.g., abelian and certain nilpotent ones. In this case the question of Steinness of X * is discussed.

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“…438] applies to show univalency of such a local action. Then by [HI,Theorem 2,p. 38] there exists a possibly non-Hausdorff universal globalization X * .…”

Section: Theorem 1 Let X Be a Taut R-manifold Then There Exists A Umentioning

confidence: 94%

Section: Existence Of Globalizationsmentioning

confidence: 99%

Section: Theorem 1 Let X Be a Taut R-manifold Then There Exists A Umentioning

confidence: 99%

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Induced local actions on taut and Stein manifolds

Iannuzzi

2003

Proc. Amer. Math. Soc.

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“…In order to study this in the framework of geometric invariant theory, it is convenient to embed X as an open K invariant subset of a good complex space X * with a holomorphic action of the reductive complexified group K C . By the results of Palais [17] and Heinzner-Iannuzzi [11], in a broad variety of situations a canonical space X * as above can be constructed as a non-necessarily Hausdorff complex space. However Heinzner proved in [7] that if X is Stein, then X * exists, is Hausdorff, and in fact it is a Stein space.…”

Section: Introductionmentioning

confidence: 99%

“…In the case X is a Kaeler manifold these weak pseudoconvexity notions are sufficient to prove vanishing theorems for cohomology of holomorphic vector bundles over X * ; see [4]. Moreover the results of [11] allow, in the hamiltonian situation, to construct suitable quotients. Note that our pseudoconvexity condition seems to be close to optimal, at least in the case of S 1 actions, to make sure that the Hausdorff property holds for every complexification X * .…”

Section: Introductionmentioning

confidence: 99%

Torus actions on weakly pseudoconvex spaces

Trapani

2005

Trans. Amer. Math. Soc.

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Kählerian Reduction in Steps

Greb

Heinzner

2009

Progress in Mathematics

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We study Hamiltonian actions of compact Lie groups K on Kähler manifolds which extend to a holomorphic action of the complexified group K C . For a closed normal subgroup L of K we show that the Kählerian reduction with respect to L is a stratified Hamiltonian Kähler K C /L C -space whose Kählerian reduction with respect to K/L is naturally isomorphic to the Kählerian reduction of the original manifold with respect to K. * The first author is supported by a Promotionsstipendium of the Studienstiftung des deutschen Volkes and by SFB/TR 12 of the DFG.† The second author is partly supported by SFB/TR 12 of the DFG.by the flow of the vector field V H associated to the Hamiltonian H via the equation dH = ı V H ω.A Hamiltonian system with symmetries is a system (X, ω, K, µ, H) where H is invariant with respect to the K-action. In this case it follows from a) thatholds for all ξ ∈ k. This implies Noether's principle in the following geometric formulation: every component µ ξ of the momentum map is a constant of motion for the system described by H.The previous considerations imply that level sets of µ are invariant under the flow of V H . In many cases questions can be organised so that µ −1 (0) is the momentum fibre of interest. By b), the group K acts on the level set µ −1 (0). Let us first consider this action on an infinitesimal level. If we fixThe optimal situation appears if µ has maximal rank at x 0 . In this case, M is smooth at x 0 and T x 0 M = ker(dµ(x 0 )) = (k • x 0 ) ⊥ω holds. It follows that M is a coisotropic K-stable submanifold of X and the symplectic formThese observations imply that once the space M/K is smooth, it will be a symplectic manifold. This is the content of the Marsden-Weinstein-Theorem (see [MW74]):If K acts freely and properly on M, the quotient M/K is a symplectic manifold whose symplectic formω is characterised via the equationHere, π : M → M/K denotes the quotient map and i M : M → X is the inclusion. Furthermore, the restriction of the K-invariant Hamiltonian H to M induces a smooth functionH on M/K. The Hamiltonian system on (M/K,ω) associated toH captures the essential (symmetryindependent) properties of the original K-invariant system that was given by H.The Marsden-Weinstein construction is natural in the sense that it can be done in steps. This means that for a normal closed subgroup L of K, the restricted momentum map µ L : X → l * is K-equivariant, the induced K-action on µ −1 L (0)/L is Hamiltonian with momentum mapμ induced by µ and the symplectic reductionμ −1 (0)/K is symplectomorphic to M/K. Removing the restrictive regularity assumptions of [MW74], it is proven in [SL91] that symplectic reduction can be carried out for general group actions of compact Lie groups yielding stratified symplectic quotient spaces M/K, i.e. stratified spaces where all strata are symplectic manifolds. The paper [HHL94] proposed an approach to this singular symplectic reduction based on embedding symplectic manifolds into Kähler manifolds and a Kähler reduction theory for Kähler manifolds. Rough...

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Integration of local actions on holomorphic fiber spaces

Cited by 15 publications

References 15 publications

Induced local actions on taut and Stein manifolds

Induced local actions on taut and Stein manifolds

Torus actions on weakly pseudoconvex spaces

Kählerian Reduction in Steps

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