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Cohomologie de Floer, hyperbolicités symplectique et pseudocmplexe.

Abstract : On one side, from the properties of Floer cohomology, invariant associated to a symplectic manifold, I define and study a notion of symplectic hyperbolicity and a symplectic capacity measuring it. On the other side, the usual notions of complex hyperbolicity can be straightforwardly generalized to the case of almost-complex manifolds by using pseudoholomorphic curves. That's why I study the links between these two notions of hyperbolicities when a manifold is provided with some compatible symplectic and almost-complex structures. I mainly explain how the non-symplectic hyperbolicity implies the existence of pseudoholomorphic curves, and so the non-complex hyperbolicity. Thanks to this analysis, I could both better understand the Floer cohomology and get new results on almost-complex hyperbolicity. I notably prove results of stability for non-complex hyperbolicity under deformation of the almost complex structure among the set of the almost-complex structures compatible with a fixed non-hyperbolic symplectic structure, thus generalizing Bangert theorem that gave this same result in the special case of the standard torus. Moreover, I tackle the issue of complex hyperbolicity of foliations: through the introduction of an invariant tensor associated to the foliation, I study the existence of foliated holomorphic cylinder.
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Submitted on : Wednesday, July 21, 2010 - 9:54:46 AM
Last modification on : Wednesday, October 20, 2021 - 12:23:55 AM
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  • HAL Id : pastel-00000702, version 1



Anne-Laure Biolley. Cohomologie de Floer, hyperbolicités symplectique et pseudocmplexe.. Mathématiques [math]. Ecole Polytechnique X, 2008. Français. ⟨pastel-00000702⟩



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