All Questions
9 questions
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 ...
- 6,025
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 ...
- 1,325
6
votes
1
answer
798
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 ...
- 1,017
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, ...
- 272
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 ...
- 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_{...
- 7,527
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^\...
- 711
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.
...
- 5,701
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 $*$ ...
- 3,019