Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems

By St'ephane Guillermou, Masaki Kashiwara, P. Schapira. 2010

125 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/029a458067d516887a71c144e10e0bff9e4a591e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Infinitely many Lagrangian fillings (2020, methodology): “t) and (4 ;4) (right). are all smoothly isotopic. We will distinguish these Lagrangian llings by studying their action on part of the coordinate ring of the moduli of framed constructible sheaves M() [34, 38, 59] for certain Legendrian links (S3;˘ st). The techniques we use for our results illustrat”
Quantization of conic Lagrangian submanifolds of cotangent bundles (2012, background): “rosupport Λu = Λ ∪T′ u(Λ) where T ′ u is the translation by uis the factor Rof M×R. We can ﬁnd aHamiltonian isotopy Φ ofT˙∗(M×R) such thatΦ s(Λ) = Λ and Φs(T′ u(Λ)) = T ′ u+s(Λ) for s,u > 0. Using [4] we can associate with Φ a sheaf on (M×R)2 whose composition with F u gives Fu+s with microsuppor”
Microlocal Morse theory of wrapped Fukaya categories (2018, background): “riant under contact isotopy of S∗M, something which is not apparent from the present combinatorial prescription. This is immediate on the Fukaya side, and on the sheaf side it is ‘sheaf quantization’ [23]. In fact, there are even stronger invariance statements: it is shown in [20] that in fact C(Λ) ”
Microlocal theory of sheaves and Tamarkin's non displaceability theorem (2011, methodology): “In Section 2 we recall the main theorem of [2] which allows one to quantize homogeneous Hamiltonian isotopies and we also give some geometrical tools relying homogeneous and non homogeneous symplectic geometry.”