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3
votes
0answers
165 views

The hypergeometric pullback conjecture

Here arXiv:math/0510287, Golishev proposed the following conjecture: The hypergeometric pullback conjecture. Let $X$ be a Fano variety. Then,for any constituent $C$ of the quantum D-module $Q$ there ...
11
votes
1answer
647 views

On the definition of regularity

In the litterature on D-modules, there are many definitions of regularity of holonomic D-modules. (1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its ...
4
votes
2answers
692 views

Confusion about D-modules and functors

Given a morphism $f:X\rightarrow Y$ between smooth complex varieties, one can define functors from the bounded derived category with holonomic cohomology on $Y$ to the same category on $X$. The ...
4
votes
1answer
311 views

Direct image of Lagrangian subspaces of the co-tangent bundle:

Let p:X \to Y be a map of smooth algebraic varieties. Let $C \subset T^\* X$ be a (locally closed) submanifold. Denote by $p_\*(C) \subset T^\* Y$ the following set: $\ \ \ \ \ \ \ \ \ \ \ \ \ \ ...
16
votes
4answers
2k views

Classification of PDE

Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for ...
11
votes
2answers
1k views

The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...
5
votes
1answer
238 views

D-modules on affine space that are regular at infinity

If I have a D-module $M$ on $\mathbb{A}^n$ (which is essentially the same thing as a module over the Weyl algebra), then I can push this D-module forward to $\mathbb{P}^n$ to get a D-module $j_*M$ on ...
5
votes
3answers
813 views

For quasi-coherent D-Modules

It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group ...
2
votes
1answer
295 views

Descriptions of projectives (injectives) in category of D-modules

Is there any work describing (indecomposable)projectives, injectives in category of D-modules on some flag variety? I wonder whether someone has used quivers(say Auslander-Reiten sequences)to ...
5
votes
0answers
307 views

Blow ups and Characteristic varieties

Let $f:Y\to X$ be a morphism of smooth varieties (complex analytic or algebraic as you wish). We have $$ T^* Y \overset{f_d}{\longleftarrow} Y\times_X T^*X \overset{f_\pi}{\longrightarrow} T^* X ...
7
votes
0answers
391 views

Status of Borho and Brylinski's irreducibility conjectures?

In Differential Operators on Homogeneous Spaces, III, Section 6.7, Borho and Brylinski make a long series of equivalent conjectures which they show are all equivalent. Perhaps the easiest statement ...
5
votes
1answer
690 views

Relation between holonomic D-modules and perverse sheaves

Given a smooth complex algebraic variety, the Riemann-Hilbert-correspondence tells us, that the category of perverse sheaves is equivalent to the category of regular, holonomic D-modules. However not ...
6
votes
1answer
474 views

How does one interpret the naive t-structure on constructible sheaves as a t-structure on D-modules?

By the Riemann-Hilbert correspondence, there is an equivalence between (1) $\mathcal{D}\operatorname{-mod}(X)$ , the (derived) category of holonomic D-modules on a complex variety X, and (2) ...
6
votes
1answer
913 views

Computation of vanishing cycles

Here's the problem I'm looking at: $F$ is a perverse sheaf (or a regular holonomic D-module, or even a mixed hodge module) on $\mathbb{C}^2$ stratified by $z_1 = 0$, $z_2=0$. It can be caracterized ...
8
votes
3answers
1k views

What is a twisted D-Module intuitively?

When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of ...
5
votes
2answers
823 views

Relation between characteristic variety and support of D-Module

Given a coherent $\mathcal{D}_X$-Module $M$, one can assign to it its characteristic variety $$ch(M)\subseteq T^*X$$ or one could look at its support $$supp(M)\subseteq X$$ as an $\mathcal {O}_X$ ...
5
votes
1answer
458 views

Reconstruction from category of D-modules on variety

Arinkin has a theorem which says that an abelian variety can be reconstructed from its derived category of coherent D-modules. D.Orlov conjectured that this theorem is true for any variety. My ...
1
vote
1answer
581 views

Is simple non-holonomic D-module a local concept?

It is well known that we can use the Riemann-Hilbert correspondence to describe holonomic D-modules in terms of a category of perverse sheaves on some variety $X$. And $U|\rightarrow Perv(U)$ is a ...
7
votes
4answers
1k views

Gluing perverse sheaves?

It might be a stupid question. How to glue perverse sheaves? I am considering the following example. Flag variety of $sl_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted ...
0
votes
2answers
507 views

What is a Module over a Lie algebroid?

Let $\alpha: \mathfrak g_A \to T_{A/k}$ be a Lie algebroid over a $k$-algebra $A$. Numerous facts about and its universal enveloping algebra comes from the theory of ring differential operators on ...
4
votes
1answer
420 views

Are these notions of strongly equivariant D-modules equivalent?

It seems that there are two notions of strongly equivaraint $D_X$- Modules and I would like to know if they are equivalent, or at least how they are related. Let $\rho: G\times X \rightarrow X$ be an ...
1
vote
3answers
240 views

TDO basic facts reference request

Hello, where can I read about some basic properties of twisted D-Modules? I would like to know, a reference, that describes how to glue these modules together/pull them back/push them forward.
6
votes
1answer
689 views

Localization of vanishing cycles

Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function. Question: Is it true that the $\lambda \in A^1$ ...
10
votes
1answer
875 views

The Infinitesimal topos in positive characteristic

This question was inspired by and is somewhat related to this question. In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie des schemas" ...
11
votes
1answer
1k views

What is the recent development of D-module and representation theory of Kac-Moody algebra?

I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me. It seems that there are several approaches to localize Kac-Moody algebra(in ...
5
votes
0answers
398 views

V-filtration of D-modules associated to a monomial

Hi In Mixed Hodge modules Saito computes the Verdier specialisation of a D-modules with respect to a monomial $g = x_1^{m_1}\ldots x_n^{m_n}$. This is a very nice result as I find such explicit ...
23
votes
1answer
2k views

D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...
3
votes
0answers
443 views

Sebastiani-Thom isomorphism for D-modules

Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$. The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes ...
7
votes
3answers
738 views

Making D-modules on affine varieties more explicit

This is a follow up to my question about D-modules supported on the nilpotent cone. I got some good answers there but now I have a more basic question. Consider an affine algebraic variety X, a ...
17
votes
3answers
2k views

How do I compare the different notions of Fourier transform for sheaves?

There is a close but not perfect relationship between algebraic D-modules on C^n, constructible sheaves on C^n in the analytic topology, and \ell-adic sheaves on an n-dimensional vector space over a ...
11
votes
3answers
995 views

D-modules supported on the nilpotent cone

I am reading some papers which involve D-modules on a Lie algebra g, which are supported on the nilpotent cone n. They are equivariant for the action of G. (In particular, I consider g=sl_n). It ...
8
votes
1answer
518 views

what is the connection between D-modules and coordinate bundles?

Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the pro-algebraic group of automorphims of $k[[x_1,...x_n]]$. Let $G_0$ be the subgroup of automorphisms preserving the closed point (note ...
7
votes
3answers
2k views

Beilinson-Bernstein and Koszul duality

For geometric representation theorists down here. Consider the Beilinson-Bernstein theorem: Functor of global sections establishes the correspondence between twisted D-modules with fixed ...