283 research outputs found
Three lectures on Algebraic Microlocal Analysis
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.Comment: small correction
Stacks of quantization-deformation modules on complex symplectic manifolds
On a complex symplectic manifold, we construct the stack of
quantization-deformation modules, that is, (twisted) modules of
microdifferential operators with an extra central parameter, a substitute to
the lack of homogeneity. We also quantize involutive submanifolds of contact
manifolds.Comment: In this new version, we have deleted from the previous one Lemma 7.6,
Proposition 7.7 and Proposition 7.8 which were erroneou
Construction of sheaves on the subanalytic site
On a real analytic manifold M, we construct the linear subanalytic
Grothendieck topology Msal together with the natural morphism of sites
from Msa to Msal, where Msa is the usual subanalytic site. Our first result is
that the derived direct image functor by 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.Comment: v 5: Chapter 5 has been expanded and other minor correction
Regular and irregular holonomic D-modules
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.Comment: 114 page
Deformation quantization modules
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.Comment: This paper develops the results of Deformation quantization modules I
(arXiv:0802.1245) and II (arXiv:0809.4309), and also contains new results and
new remarks. It will appear in Asterisque, Soc. Math. France (2012
Irregular holonomic kernels and Laplace transform
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.Comment: 62 pages. 2nd version typoes correcte
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