2004/04/30 by Anne-Laure Biolley, Biolley, Anne-Laure
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.CV #math.SG #msc:32Q60 #msc:32Q65 #msc:32Q68 #msc:53D40
paper · pdf · doi:10.48550/arxiv.math/0404551
45 pages
arxiv created 2004/04/30 · arxiv updated 2009/12/01
On one side, from the properties of Floer cohomology, invariant associated to a symplectic manifold, we 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 pseudo-holomorphic curves. That's why we study the links between these two notions of hyperbolicities when a manifold is provided with some compatible symplectic and almost-complex structures. We mainly explain how the non-symplectic hyperbolicity implies the existence of pseudo-holomorphic curves, and so the non-complex hyperbolicity. Thanks to this analysis, we could both better understand the Floer cohomology and get new results on almost-complex hyperbolicity. We 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.