Astérisque - Numéro 440 : Sheaves and symplectic geometry of cotangent bundles (Revue)

The aim of this paper is to apply the microlocal theory of sheaves of Kashiwara-Schapira to the symplectic geometry of cotangent bundles, following ideas of Nadler-Zaslow and Tamarkin. We recall the main notions and results of the microlocal theory of sheaves, in particular the microsupport of sheaves. The microsupport of a sheaf F on a manifold M is a closed conic subset of the cotangent bundle T*M which indicates in which directions we can modify a given open subset of M without modifying the cohomology of F on this subset. An important theorem of Kashiwara-Schapira says that the microsupport is coisotropic and recent works of Nadler-Zaslow and Tamarkin study in the other direction the sheaves which have for microsupport a given Lagrangian submanifold A, obtaining information on A in this way. Nadler and Zaslow made the link with the Fukaya category but Tamarkin only made use of the microlocal sheaf theory. We go on in this direction and recover several results of symplectic geometry with the help of sheaves. In particular we explain how we can recover the Gromov nonsqueezing theorem, the Gromov- Eliashberg rigidity theorem, the existence of graph selectors. We also prove a three cusps conjecture of Arnol'd about curves on the sphere. In the last sections we recover more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles.