Andrea D'Agnolo

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.

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.

Radon and Fourier transforms for D-modules

The Fourier and Radon hyperplane transforms are closely related, and one such relation was established by Brylinski [4] in the framework of holonomic D-modules. The integral kernel of the Radon hyperplane transform is associated with the hypersurface SCP P of pairs ðx; yÞ; where x is a point in the n-dimensional complex projective space P belonging to the hyperplane yAP : As it turns out, a useful variant is obtained by considering the integral transform associated with the open complement U of S in P P : In the first part of this paper, we generalize Brylinski’s result in order to encompass this variant of the Radon transform, and also to treat arbitrary quasi-coherent D-modules, as well as (twisted) abelian sheaves. Our proof is entirely geometrical, and consists in a reduction to the onedimensional case by the use of homogeneous coordinates. The second part of this paper applies the above result to the quantization of the Radon transform, in the sense of [7]. First we deal with line bundles. More precisely, let P 1⁄4 PðVÞ be the projective space of lines in the vector spaceV; denote by ð Þ 3 D R

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.

Representation Theory and Complex Analysis

A collection of advanced articles in Complex Analysis, Lie Groups, Unitary Representations and Quantum Computing, wirtten by the scientific leaders in these areas.

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

On the Laplace Transform for Tempered Holomorphic Functions

X

Morita Classes of Microdifferential Algebroids

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

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

X

On Dwork cohomology and algebraic D-modules

. To state it precisely, one needs the notion of algebroid stack, introduced by Kontsevich. In particular, the stack of WKB modules over