Stéphane Guillermou

Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems

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.

Microlocal theory of sheaves and Tamarkin's non displaceability theorem

This paper is an attempt to better understand Tamarkins approach of classical non-displaceability theorems of symplectic geometry, based on the microlocal theory of sheaves, a theory whose main features we recall here. If the main theorems are due to Tamarkin, our proofs may be rather different and in the course of the paper we introduce some new notions and obtain new results which may be of interest.

Regular holonomic D[[h]]-modules

We describe the category of regular holonomic modules over the ring D[[h]] of linear differential operators with a formal parameter h. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional t-structure related to h-torsion.

Equivariant derived category of a complete symmetric variety

Let G be a complex algebraic semi-simple adjoint group and X a smooth complete symmetric G-variety. Let L_i be the irreducible G-equivariant intersection cohomology complexes on X, and L the direct sum of the L_i. Let E= Ext(L,L) be the extension algebra of L, computed in the G-equivariant derived category of X. We considered E as a dg-algebra with differential d=0, and the E_i = Ext(L,L_i) as E-dg-modules. We show that the bounded equivariant derived category of sheaves of C-vector spaces on X is equivalent to the subcategory of the derived category of E-dg-modules generated by the E_i.

Regular holonomic

We describe the category of regular holonomic modules over the ring D[[h]] of linear differential operators with a formal parameter h. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional t-structure related to h-torsion.