Pierre Schapira

Operations on constructible functions

Abstract A constructible function ϕ on a real analytic manifold X is a Z -valued function such that the partition X=⨆ mϵZ ϕ -1 (m) is a subanalytic stratification. Here, we define new operations on constructible functions (inverse or direct images, duality) and prove some theorems related to these operations (e.g., duality commutes to direct image). As an application we solve the convolution equation ϕ ∗ ψ = α , when ϕ is the characteristic function of a convex compact set. This problem which seems to have some utility in robotics, was first considered by Guibas et al, and this paper may be considered as a new approach, and an extension to higher dimension of the material contained in their paper.

Leray’s quantization of projective duality

2 Review on correspondences for sheaves and D-modules 4 2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 D-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 E-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Correspondences for sheaves and D-modules . . . . . . . . . . . . . . 6

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.

Quantization of complex Lagrangian submanifolds

Let be a smooth Lagrangian submanifold of a complex symplectic manifold X. We construct twisted simple holonomic modules along in the stack of deformationquantization modules on X.

Conditions de positivité dans une variété symplectique complexe . Applications à l'étude des microfonctions

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1981, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Constructible functions, Lagrangian cycles and computational geometry

Given two subsets A and B of the Euclidian space IRn, how to recognize that A is contained in B ? This apparently easy problem is basic in computational geometry, and does not have (to our knowledge) a completely satisfactory answer.

Microfunctions for Boundary Value Problems

Publisher Summary This chapter discusses the microfunctions for boundary value problems. It defines a wavefront set at the boundary of convex open subsets of Rn. It also provides an overview of the application of diffraction.

Propagation at the Boundary of Analytic Singularities

We study non elliptic boundary value problems in the framework of microfunctions. We introduce the concept of N-regularity and apply it to the problem of propagation of analytic singularities at the boundary with some possible diffraction.

An inverse image theorem for sheaves with applications to the Cauchy problem

(i) In 1976 Hamada, Leray and Wagschal solved the initial value problem for a linear partial differential equation when the data are ramified along the characteristic hypersurfaces. Their proof of this result relies essentially on the precise Cauchy-Kowalevski theorem of Leray (cf [H-L-W]). (ii) In 1978 Kashiwara and Schapira proposed a new proof and an extension of the previous work to general systems. This time microdifferential operators and complex contact transformations were involved (cf [K-S 1]). (iii) In 1988 Schiltz showed how the holomorphic solution for the Cauchy problem can be expressed as a sum of functions which are holomorphic in domains whose boundary is given by the real characteristic hypersurfaces issued from the boundary of a strictly pseudoconvex domain where the data are defined (cf [Sc]).