arXiv subjects
Publications and source records attributed to Pierre Schapira.
This is a survey paper based on a series of lectures given at the IHES in February/March 2015. In a first part, we recall the main results on the tempered holomorphic solutions of D-modules in the language of indsheaves and, as an application, the Riemann-Hilbert correspondence for regular holonomic modules. In a second part, we present the enhanced version of the first part, treating along the same lines the irregular holonomic case.
Given a (not necessarily regular) holonomic D-module defined on the product of two complex manifolds, we prove that the associated correspondence commutes (in some sense) with the De Rham functor. We apply this result to the study of the classical Laplace transform. The main tools used here are the theory of ind-sheaves and its enhanced version.
These three lectures present some fundamental and classical aspects of microlocal analysis. Starting with the Sato's microlocalization functor and the microsupport of sheaves, we then construct a microlocal analogue of the Hochschild homology for sheaves and apply it to recover index theorems for D-modules and elliptic pairs. In the third lecture, we construct the ind-sheaves of temperate and Whitney holomorphic functions and give some applications to the study of irregular holonomic D-modules.
For an abelian category C and a filtrant preordered set Lambda, we prove that the derived category of the quasi-abelian category of filtered objects in C indexed by Lambda is equivalent to the derived category of the abelian category of functors from Lambda to C. We apply this result to the study of the category of filtered modules over a filtered ring in a tensor category.
This is essentially a survey paper in which we solve the global Cauchy problem on causal manifolds for hyperbolic systems of linear partial differential equations in the framework of hyperfunctions. Besides the classical Cauchy-Kowalevsky theorem, our proofs only use tools from the microlocal theory of sheaves, that is, tools of purely algebraic and geometric nature.
We define the notion of a trace kernel on a manifold M. Roughly speaking, it is a sheaf on M x M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class to such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle over M and we prove that this class is functorial with respect to the composition of kernels. This generalizes, unifies and simplifies various results of (relative) index theorems for constructible sheaves, D-modules and elliptic pairs.
This paper is an attempt to better understand Tamarkin's 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.
We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class commutes with composition of kernels, a kind of Riemann-Roch theorem in the non-commutative setting. Finally we study holonomic modules on complex symplectic manifolds and we prove in particular a constructibility theorem.
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.
We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing complexes in this framework and show that the convolution of kernels commutes with duality.
This paper is the continuation of arXiv:0802.1245. We construct the Hochschild class for coherent modules over a deformation quantization algebroid on a complex Poisson manifold. We also define the convolution of Hochschild homologies, and prove that the Hochschild class of the convolution of two coherent modules is the convolution of their Hochschild classes. We study with some details the case of symplectic deformations.
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.
This paper is a brief overview of the main contributions of Masaki Kashiwara in the domain of microlocal and algebraic analysis.
We study modules over the algebroid stack $\W[\stx]$ of deformation quantization on a complex symplectic manifold $\stx$ and recall some results: construction of an algebra for $\star$-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regular holonomic $\W[\stx]$-modules, finiteness and duality for the composition of ``good'' kernels. As a corollary, we get that the derived category of good $\W[\stx]$-modules with compact support is a Calabi-Yau category. We also give a conjectural Riemann-Roch type formula in this framework.
Consider a complex symplectic manifold $X$ and the algebroid $W_X$ of quantization-deformation. For two regular holonomic modules $L_i$ ($i=0,1$) supported by smooth Lagrangian manifolds, we prove that the complex $Rhom_{W_X}(L_1,L_0)$ is constructible and perverse and dual to the complex $Rhom_{W_X}(L_0,L_1)$.
Let $Λ$ be a smooth Lagrangian submanifold of a complex symplectic manifold $X$. We construct twisted simple holonomic modules along $Λ$ in the stack of deformation-quantization modules on $X$.
The cotangent bundle $T^*X$ to a complex manifold $X$ is classically endowed with the sheaf of $\cor$-algebras $\W[T^*X]$ of deformation quantization, where $\cor\eqdot \W[\rmptt]$ is a subfield of $\C[[\hbar,\opb{\hbar}]$. Here, we construct a new sheaf of $\cor$-algebras $\TW[T^*X]$ which contains $\W[T^*X]$ as a subalgebra and an extra central parameter $t$. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If $P$ is any section of order zero of $\W[T^*X]$, we show that $\exp(t\opb{\hbar} P)$ is well defined in $\TW[T^*X]$.
Let X be a C-infinity manifold. We construct a microlocalization functor $μ_X$ from the derived category of bounded complexes of ind-sheaves on X to the one on the cotangent bundle of X. This functor generalizes the classical theory of microlocalization.