arXiv subjects
Publications and source records attributed to Pierre Schapira.
On a spacetime endowed with a time function (such as a globally hyperbolic manifold) we solve the global Cauchy problem for hyperbolic D-modules in the framework of Sato's hyperfunctions, illustrating the effectiveness of microlocal-sheaf theoretic methods in linear problems.
A causal manifold $(M,γ)$ is a manifold $M$ endowed with a closed proper cone $γ$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of $γ$. Let $λ$ be the antipodal of the polar cone of $γ$. An open set $U$ of $M$ is called $γ$-open if its Whitney normal cone contains the interior of $γ$. Similarly, $U$ is called $λ$-open if the micro-support of the constant sheaf on $U$ is contained in $λ$. We begin by proving that the two notions coincide. Next, we prove that if $(M,γ)$ admits a ``future time function'' the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $λ$ and the derived category of sheaves on the manifold $M$ endowed with the $γ$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space.
We prove that various morphisms related to the six Grothendieck operations on sheaves become isomorphisms when restricted to (weakly) constructible sheaves. To this end, we first study some properties of weakly cohomologically constructible sheaves. We then deduce several compatibilities of the six operations in the context of (weakly) $\mathbb{R}$-constructible sheaves.
This paper, to appear in the ``Notices of the AMS'' 2024, is a modified version of a text already appeared in this journal, Feb. 2007 after a first publication in French, in ``La Gazette des Math{é}maticiens'' 97 (2003) on the occasion of Sato's reception of the 2002/2003 Wolf prize.
We introduce the category of b-analytic manifolds, a natural tool to define constructible sheaves and functions up to infinity. We study with some details the operations on these objects and also recall the Radon transform for constructible functions.
A critical essay of the book ``Récoltes et Semailles'' by Alexander Grothendieck, recently published by Gallimard editions.
Given a topological space $X$, a thickening kernel is a monoidal presheaf on $(\mathbb{R}_{\geq0},+)$ with values in the monoidal category of derived kernels on $X$. A bi-thickening kernel is defined on $(\mathbb{R},+)$. To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on $X$. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing $0$, allowing us to construct (bi-)thickenings in two different situations. First, when $X$ is a ``good'' metric space, starting with small usual thickenings of the diagonal. The associated interleaving distance satisfies the stability property and Lipschitz kernels give rise to Lipschitz maps. Second, by using [GKS12], when $X$ is a manifold and one is given a non-positive Hamiltonian isotopy on the cotangent bundle. In case $X$ is a complete Riemannian manifold having a strictly positive convexity radius, we prove that it is a good metric space and that the two bi-thickening kernels of the diagonal, one associated with the distance, the other with the geodesic flow, coincide.
Let $X$ be a real analytic manifold endowed with a distance satisfying suitable properties and let $\mathbf{k}$ be a field. In [PS20], the authors construct a pseudo-distance on the derived category of sheaves of $\mathbf{k}$-modules on $X$, generalizing a previous construction of [KS18]. Here, we prove that if the distance between two constructible sheaves with compact support (or more generally, constructible sheaves up to infinity) on $X$ is zero, then these two sheaves are isomorphic. This answers in particular a question of [KS18].
Consider a complex Stein manifold X and a subanalytic relatively compact Stein open subset U of X.. We prove the vanishing on U of the holomorphic temperate cohomology.
On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.
On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL gamma-sheaves, that is, PL sheaves associated with the gamma-topology, for a closed convex polyhedral proper cone gamma. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.
An overview on Masaki Kashiwara's mathematical work since 1970
We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space V. By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images. Then we assume that V is endowed with a closed convex proper cone $γ$ with non empty interior and study $γ$-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the $γ$-topology). We prove that such sheaves may be approximated (for the pseudo-distance) by "piecewise linear" $γ$-sheaves. Finally we show that these last sheaves are constant on stratifications by $γ$-locally closed sets, an analogue of barcodes in higher dimension.
We extend the classical Wick rotation to D-modules and higher codimensional submanifolds.
We shall explain how the idea of microlocal analysis of the seventies has been reformulated in the framework of sheaf theory in the eighties and then applied to various branches of mathematics, such as linear partial differential equations or symplectic topology.
Microlocal sheaf theory of \cite{KS90} makes an essential use of an extension lemma for sheaves due to Kashiwara, and this lemma is based on a criterion of the same author giving conditions in order that a functor defined in $\mathbb{R}$ with values in the category $Sets$ of sets be constant. In a first part of this paper, using classical tools, we show how to generalize the extension lemma to the case of the unbounded derived category. In a second part, we extend Kashiwara's result on constant functors by replacing the category $Sets$ with the $\infty$-category of spaces and apply it to generalize the extension lemma to $\infty$-sheaves, the $\infty$-categorical version of sheaves. Finally, we define the micro-support of sheaves with values in a stable $(\infty,1)$-category.
On a real analytic manifold M, we construct the linear subanalytic Grothendieck topology Msal together with the natural morphism of sites $ρ$ from Msa to Msal, where Msa is the usual subanalytic site. Our first result is that the derived direct image functor by $ρ$ admits a right adjoint, allowing us to associate functorially a sheaf (in the derived sense) on Msa to a presheaf on Msa satisfying suitable properties, this sheaf having the same sections that the presheaf on any open set with Lipschitz boundary. We apply this construction to various presheaves on real manifolds, such as the presheaves of functions with temperate growth of a given order at the boundary or with Gevrey growth at the boundary. On a complex manifold endowed with the subanalytic topology, the Dolbeault complexes associated with these new sheaves allow us to obtain various sheaves of holomorphic functions with growth. As an application, we can endow functorially regular holonomic D-modules with a filtration, in the derived sense.
We introduce a class of causal manifolds which contains the globally hyperbolic spacetimes and we prove global propagation theorems for sheaves on such manifolds. As an application, we solve globally the Cauchy problem for hyperfunction solutions of hyperbolic systems.