1,330 research outputs found
A microlocal approach to the enhanced Fourier-Sato transform in dimension one
Let be a holonomic algebraic -module on the affine
line. Its exponential factors are Puiseux germs describing the growth of
holomorphic solutions to at irregular points. The stationary
phase formula states that the exponential factors of the Fourier transform of
are obtained by Legendre transform from the exponential factors
of . We give a microlocal proof of this fact, by translating it in
terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.Comment: 56 page
Riemann-Hilbert correspondence for holonomic D-modules
The classical Riemann-Hilbert correspondence establishes an equivalence
between the triangulated category of regular holonomic D-modules and that of
constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence
for holonomic D-modules which are not necessarily regular. The construction of
our target category is based on the theory of ind-sheaves by Kashiwara-Schapira
and influenced by Tamarkin's work. Among the main ingredients of our proof is
the description of the structure of flat meromorphic connections due to
Mochizuki and Kedlaya.Comment: 114pages; v.2 minor changes, 114 p
On Dwork cohomology and algebraic D-modules
After works by Katz, Monsky, and Adolphson-Sperber, a comparison theorem
between relative de Rham cohomology and Dwork cohomology is established in a
paper by Dimca-Maaref-Sabbah-Saito in the framework of algebraic D-modules. We
propose here an alternative proof of this result. The use of Fourier transform
techniques makes our approach more functorial.Comment: latex, 8 page
Deformation-Quantization of Complex Involutive Submanifolds
The sheaf of rings of WKB operators provides a deformation-quantization of
the cotangent bundle to a complex manifold. On a complex symplectic manifold
there may not exist a sheaf of rings locally isomorphic to a ring of WKB
operators. The idea is then to consider the whole family of locally defined
sheaves of WKB operators as the deformation-quantization of . To state it
precisely, one needs the notion of algebroid stack, introduced by Kontsevich.
In particular, the stack of WKB modules over defined in Polesello-Schapira
(see also Kashiwara for the contact case) is better understood as the stack of
modules over the algebroid stack of deformation-quantization of .
Let be an involutive submanifold of , and assume for simplicity that
the quotient of by its bicharacteristic leaves is isomorphic to a complex
symplectic manifold . The algebra of endomorphisms of a simple WKB module
along is locally (anti-)isomorphic to the pull-back of WKB operators on
. Hence we may say that a simple module provides a deformation-quantization
of . Again, since in general there do not exist globally defined simple WKB
modules, the idea is to consider the algebroid stack of locally defined simple
WKB modules as the deformation-quantization of .
In this paper we start by defining what an algebroid stack is, and how it is
locally described. We then discuss the algebroid stack of WKB operators on a
complex symplectic manifold , and define the deformation-quantization of an
involutive submanifold by means of simple WKB modules along . Finally,
we relate this deformation-quantization to that given by WKB operators on the
quotient of by its bicharacteristic leaves.Comment: 11 page
Global propagation on causal manifolds
The micro-support of sheaves is a tool to describe local propagation results.
A natural problem is then to give sufficient conditions to get global
propagation results from the knowledge of the micro-support. This is the aim of
this paper. As an application, we consider the problem of global existence for
solutions to hyperbolic systems (in the hyperfunction and distribution case),
along the lines of Leray. Causal manifolds, and in particular homogeneous
causal manifolds as considered by Faraut et al., give examples of manifolds to
which our results apply.Comment: 15 pages, LaTeX, requires XYpi
A note on quantization of complex symplectic manifolds
To a complex symplectic manifold X we associate a canonical quantization
algebroid. Our construction is similar to that of Polesello-Schapira's
deformation-quantization algebroid, but the deformation parameter is no longer
central. If X is compact, the triangulated category of regular holonomic
quantization modules is a linear complex Calabi-Yau category of dimension 1 +
dim X.Comment: Statement of Theorem 3.8 correcte
Stacks of twisted modules and integral transforms
Stacks were introduced by Grothendieck and Giraud and are, roughly speaking,
sheaves of categories. Kashiwara developed the theory of twisted modules, which
are objects of stacks locally equivalent to stacks of modules over sheaves of
rings. In this paper we recall these notions, and we develop the formalism of
operations for stacks of twisted modules. As an application, we state a twisted
version of an adjunction formula which is of use in the theory of integral
transforms for sheaves and D-modules.Comment: latex, 43 page
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.Comment: 39 page
On the Laplace transform for tempered holomorphic functions
In order to discuss the Fourier-Sato transform of not necessarily conic
sheaves, we compensate the lack of homogeneity by adding an extra variable. We
can then obtain Paley-Wiener type results, using a theorem by Kashiwara and
Schapira on the Laplace transform for tempered holomorphic functions. As a key
tool in our approach, we introduce the subanalytic sheaf of holomorphic
functions with exponential growth, which should be of independent interest.Comment: 31 pages, section numbering modified to reflect the published version
of the pape
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