Rigidity of Hamiltonian actions on Poisson manifolds

@article{Miranda2012RigidityOH,
  title={Rigidity of Hamiltonian actions on Poisson manifolds},
  author={Eva Miranda and Philippe Monnier and Nguyen Tien Zung},
  journal={Advances in Mathematics},
  year={2012},
  volume={229},
  pages={1136-1179}
}
Abstract This paper is about the rigidity of compact group actions in the Poisson context. The main result is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash–Moser normal form theorem for closed subgroups of SCI type. This Nash–Moser normal form has other applications to stability results that we will explore in a future paper. We also review some classical rigidity results for differentiable actions of compact Lie groups and export it to the case of… Expand
RIGIDITY OF POISSON LIE GROUP ACTIONS
In this paper we prove that close infinitesimal momentum maps associated to Poisson Lie actions are equivalent under some mild assumptions. We also obtain rigidity theorems for actual momentum mapsExpand
Rigidity of infinitesimal momentum maps
In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions areExpand
Symplectic and Poisson structures with symmetries in interaction
Hamiltonian actions constitute a central object of study in symplectic geometry. Special attention has been devoted to the toric case. Toric symplectic manifolds provide natural examples ofExpand
A note on symplectic and Poisson linearization of semisimple Lie algebra actions
In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization forExpand
Coupling symmetries with Poisson structures
We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact LieExpand
Rigidity of actions on presymplectic manifolds
We prove the rigidity of presymplectic actions of a compact semisimple Lie algebra on a presymplectic manifold of constant rank in the local and global case. The proof uses an abstract normal formExpand
Rigidity around Poisson submanifolds
We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies aExpand
Rigidity of cotangent lifts and integrable systems
Abstract In this article we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplecticExpand
Local analytic geometry of generalized complex structures
A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper, weExpand
Normal forms in Poisson geometry
The structure of Poisson manifolds is highly nontrivial even locally. The first important result in this direction is Conn's linearization theorem around fixed points. One of the main results of thisExpand
...
1
2
3
...

References

SHOWING 1-10 OF 35 REFERENCES
Lectures on Symplectic Manifolds
Introduction Symplectic manifolds and lagrangian submanifolds, examples Lagrangian splittings, real and complex polarizations, Kahler manifolds Reduction, the calculus of canonical relations,Expand
A note on equivariant normal forms of Poisson structures
We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization resultExpand
Integrability of Lie brackets
In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by twoExpand
MOMENTUM MAPPINGS AND POISSON COHOMOLOGY
We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentumExpand
Levi decomposition for smooth Poisson structures
We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. This Levi decomposition is a kind of normal form or partialExpand
Integrability of Poisson Brackets
We discuss the integration of Poisson brackets, motivated by our recent solution to the integrability problem for general Lie brackets. We give the precise obstructions to integrating PoissonExpand
Compact Groups of Differentiable Transformation
H. Cartan' has investigated automorphisms and general transformations into itself of a domain in several complex variables by means of complex analytic functions. His main results are as follows: (i)Expand
Lie-Poisson structure on some Poisson Lie groups
Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group isExpand
A singular Poincaré lemma
We prove a Poincare lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a CartanExpand
Cohomology Theory of Lie Groups and Lie Algebras
The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced toExpand
...
1
2
3
4
...