Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems

TL;DR

This paper develops a sheaf-theoretic framework for Hamiltonian isotopies on cotangent bundles, proving existence and uniqueness of certain sheaves, and applies these results to non-displaceability problems in symplectic and contact topology.

Contribution

It introduces a sheaf quantization approach for Hamiltonian isotopies, extending non-displaceability results and stability of Morse inequalities under isotopies.

Findings

Existence and uniqueness of a sheaf K associated with Hamiltonian isotopies.

Stability of Morse inequalities under Hamiltonian isotopies.

Non-displaceability results for positive isotopies in contact topology.

Abstract

Let I be an open interval, M be a real manifold, T*M its cotangent bundle and \Phi={\phi_t}, t in I, a homogeneous Hamiltonian isotopy of T*M defined outside the zero-section. Let \Lambda be the conic Lagrangian submanifold associated with \Phi (\Lambda is a subset of T*M x T*M x T*I). We prove the existence and unicity of a sheaf K on MxMxI whose microsupport is contained in the union of \Lambda and the zero-section and whose restriction to t=0 is the constant sheaf on the diagonal of MxM. We give applications of this result to problems of non displaceability in contact and symplectic topology. In particular we prove that some strong Morse inequalities are stable by Hamiltonian isotopies and we also give results of non displaceability for positive isotopies in the contact setting. In this new version we suppress one hypothesis in the main theorem and we extend the result of non displaceability for positive isotopies.