Rigidity around Poisson submanifolds

@article{Marcut2014RigidityAP,
  title={Rigidity around Poisson submanifolds},
  author={I. Marcut},
  journal={Acta Mathematica},
  year={2014},
  volume={213},
  pages={137-198}
}
  • I. Marcut
  • Published 2014
  • Mathematics
  • Acta Mathematica
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 a stronger version of Conn’s linearization theorem [2], also proving that Conn’s theorem is a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem [7]. We can also use the rigidity theorem to… Expand
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References

SHOWING 1-10 OF 29 REFERENCES
Formal equivalence of Poisson structures around Poisson submanifolds
Let (M,π) be a Poisson manifold. A Poisson submanifold P ⊂ M gives rise to a Lie algebroid AP → P. Formal deformations of π around P are controlled by certain cohomology groups associated to AP.Expand
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
Rigidity of Hamiltonian actions on Poisson manifolds
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 aExpand
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
Coupling Tensors and Poisson Geometry near a Single Symplectic Leaf
In the framework of the connection theory, a contravariant analog of the Sternberg coupling procedure is developed for studying a natural class of Poisson structures on fiber bundles, called couplingExpand
Geometry of orbit spaces of proper Lie groupoids
In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for properExpand
Extensions of symplectic groupoids and quantization.
An important role of Poisson manifolds is äs intermediate objects between ordinary manifolds, with their commutative algebras of functions, and the "noncommutative spaces" of quantum mechanics. Up toExpand
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
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
A normal form theorem around symplectic leaves
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearizationExpand
...
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