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2 votes
0 answers
126 views

Compact generators of $\mathcal{D}\text{-Mod}$ via Mayer Vietoris

Let $X$ be a complex variety (or any space for whom a category of sheaves with the six functors is defined), and $$i\ :\ Z\ \to\ X\ \leftarrow\ U\ :\ j$$ be complementary open and closed embeddings. ...
Pulcinella's user avatar
  • 5,701
2 votes
1 answer
218 views

About the support of a holonomic D-module

Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b_\text{h}(\mathcal{D}_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\...
Gabriel's user avatar
  • 721
1 vote
0 answers
172 views

When does a $D$-module think it’s a pullback along a smooth morphism?

Let $X$ and $Y$ be two algebraic varieties, and let $f: X \to Y$ be a morphism. Suppose $A$ is a holonomic $D$-module on $Y$. In this situation we can pull $A$ back to $X$ using either the $!$ or $*$ ...
Exit path's user avatar
  • 3,019
6 votes
1 answer
2k views

How to understand the Fourier-Sato transform and microlocalization functors?

Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, ...
Mathmank's user avatar
  • 272
9 votes
1 answer
1k views

Kozsul resolution of $\mathcal{O}_X$

Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free ...
Federico Barbacovi's user avatar
24 votes
1 answer
837 views

Is there a useful theory of D-modules on smooth (non-analytic) manifolds?

D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
ಠ_ಠ's user avatar
  • 6,025
6 votes
1 answer
799 views

Example of non-holonomic D-module and explicit computation of characteristic variety

I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know ...
C. Dubussy's user avatar
  • 1,017
4 votes
1 answer
511 views

Nearby cycles and specialisation - properties

I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
Sasha's user avatar
  • 5,562
3 votes
0 answers
260 views

Pull back of D-modules and Koszul resolution

Consider an holonomic D-module on a smooth algebraic variety $X$ over a field $k$ of caracteristic 0. Let $i: Y \hookrightarrow X$ be a regular embedding. $Li^* M = \mathcal{D}_{Y\to X} \otimes^L_{...
AFK's user avatar
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