Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems

2010/05/31 by Stephane Guillermou, Masaki Kashiwara, Pierre Schapira · 5 citations
Mathematics · #math.SG #math.AT

paper · pdf · doi:10.1215/00127094-1507367

published as Duke Math. J. 161, no. 2 (2012), 201-245 · Corrected version, to appear at "Duke Math. Journal"

arxiv created 2011/06/22 · arxiv updated 2019/12/19

Abstract

Let I be an open interval, M be a real manifold, T*M its cotangent bundle and Φ=ϕ_t, t in I, a homogeneous Hamiltonian isotopy of T*M defined outside the zero-section. Let Λbe the conic Lagrangian submanifold associated with Φ(Λis 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 Λand 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.

Cited by

• Topological constraints on clean Lagrangian intersections via microlocal sheaf theory
• Mirror symmetry for the Painlevé character varieties
• Density of fibers for the filtered Fukaya category of T^*N
• A topological classification of generating functions
• Derived sheaves in locally conformally symplectic geometry

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• Quantization of conic Lagrangian submanifolds of cotangent bundles
• Derived sheaves in locally conformally symplectic geometry
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• A proof of the Hamiltonian Thom Isotopy Lemma