2
votes
1answer
105 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 ...
2
votes
1answer
92 views

Sabbah b-functions factoring

Sabbah defined a version of b-functions for multiple functions in his 1987 paper here: http://archive.numdam.org/ARCHIVE/CM/CM_1987__62_3/CM_1987__62_3_283_0/CM_1987__62_3_283_0.pdf According to ...
4
votes
1answer
182 views

Smooth mixed hodge modules - representations of fundamental group?

I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...
2
votes
0answers
93 views

Algebraic methods in pde [duplicate]

I'm finding myself with a linear, very symetric system of first orders pde with polynomial coefficients. Wandering on the web, i learnt there is some nice alebraic way to deal with it involving ...
11
votes
2answers
444 views

Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
1
vote
1answer
144 views

Most general “finiteness of de Rham cohomology” statement for holonomic $D$-modules in the algebraic case?

Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) ...
1
vote
0answers
80 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} ...
4
votes
0answers
172 views

singular support of D-module smooth w.r.t. a stratification

(1) Suppose that $X$ is a smooth complex algebraic variety, stratified by some nice smooth stratification $S$. Let $M$ be a $D$-module on $X$, s.t. its shriek-pullback (or star... whatever is ...
2
votes
2answers
287 views

About “de-Rham” and “l-adic” local systems - comparison

Hello, Suppose that $k$ is an algebraically closed field of char. 0. Let $X$ be a smooth connected variety over $k$. Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. ...
2
votes
1answer
238 views

D-affine morphisms and composition

I am just a beginner in $D$-module, so this could be a stupid question, but I can't find an easy reference for it. I would like to define the notion of $D$-affine morphism. The most obvious way would ...
1
vote
0answers
214 views

Weight decomposition and eigenspaces Euler vector field

Let $V$ be a finite dimensional vector space over $\mathbb{C}$, denote $V^{\times} = V -0$ and let $\pi : V^{\times} \rightarrow \mathbb{P}(V)$ be the quotient by the natural action of ...
4
votes
1answer
435 views

Analogues of D-modules and constructible sheaves

For a smooth complex variety, one can consider the category of say holonomic $\mathcal D$ modules on it. It is equipped with the deRham functor, which turns a $\cal D$-module into a constructible ...
6
votes
2answers
335 views

D-module that is coherent as O-module

Suppose that $X$ is an algebraic variety over $\mathbb C$, not necessarily smooth. Is it still true that each $\mathcal D_X$-module ($\mathcal D_X$ is of course the sheaf of differential operators) ...
1
vote
0answers
163 views

Non-characteristic is to pullback as (blank) is to pushforward.

Suppose $f:X\to Y$ is a map of smooth complex algebraic varieties. There is a pushforward functor $f_\ast : D(X) \to D(Y)$ on the derived category of $D$-modules. This certainly does not preserve ...
4
votes
1answer
282 views

Tensor product of $\mathcal{D}$-modules and constructible sheaves

The Riemann-Hilbert correspondence, as proved by Kashiwara and Mebkhout, says that for X a smooth algebraic variety over $\mathbb{C}$ there is an equivalence of triangulated categories ...
1
vote
0answers
355 views

An equivalence between $(\mathcal{D}_X^m)-\text{mod}$ and $(\mathcal{D}_X^{m+1})-\text{mod}$

This question is related to my other question. Consider a scheme $X$ over $S=\text{Spec}(\mathbb{k})$ where $\mathbb{k}=\overline{\mathbb{F}_p})$; let $F: X \rightarrow X$ be the Frobenius $p$-th ...
5
votes
1answer
600 views

$\mathcal{D}$-modules of level m

My question is regarding the definition of $\mathcal{D}$-modules of level $m$ given in this paper. As an example, let $X=\mathbb{A}^1$ over $S=\text{Spec }\overline{\mathbb{F}_p}$; I was told that a ...
4
votes
0answers
208 views

$D$-modules and quasi-projective varieties?

Until recently I was under the impression, that for any morphism $f:X\rightarrow Y$ of smooth complex varieties there exist functors six functors $f^*,f_{*},...$ between the derived categories of ...
11
votes
1answer
598 views

$\mathcal{D}$-quasi-isomorphisms and coherent $\Omega$-modules

Let $X$ be a smooth $\mathbf{C}$-scheme of finite type. In section 7.2 of their preprint "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld define an ...
6
votes
1answer
222 views

Distinguished triangle of closed - open partition, for D-modules

Hello, I am sorry if this question is not appropriate for MO. Suppose $X$ is the affine line, $i:Z\to X$ is the origin, and $j: U \to X$ is the complement to $Z$ in $X$. I then have a distinguished ...
5
votes
0answers
160 views

Computing the constant D-module on an intersection

Let $M$ be a smooth variety, say over the complex numbers, and let $i:W \hookrightarrow M, j: Z \hookrightarrow M$ be smooth subvarieties. Let $i_{+},j_{+}$ denote (derived) pushforward of D-modules, ...
4
votes
1answer
425 views

Localizability of differential operators a la Grothendieck

Hello, Maybe this question is trivial, so sorry Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1). Then we can define the module of differential operators $D^{\leq n} ...
6
votes
1answer
386 views

l-adic Turrittin

What would be an l-adic analogue of the Turrittin-Levelt decomposition theorem? Turrittin-Levelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a ...
2
votes
1answer
463 views

Is base affine space a trivial fibration?

I am very confused about the base affine space. Let $G$ be a complex reductive algebraic group and $H=B/U$ the abstract torus. If I understand it correctly (Edit: which turns out not to be the case! ...
12
votes
3answers
858 views

Examples for D-affine varieties?

A variety is called $\mathcal D$- affine if the global section functor induces an equivalence between quasi-coherent $\mathcal D$-modules and modules over $\Gamma(X,\mathcal D)$. It is easy to see ...
12
votes
2answers
876 views

D-modules and Algebraic Solutions of PDEs

I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When ...
3
votes
1answer
326 views

Base change for the Gauss-Manin sheaf

I want to see the following thing: $\ \ $If $X$ is a smooth geometrically connected scheme over a field $k$ of characteristic 0, $U\subseteq X$ is a non-empty open, $(E,\nabla)$ is an integrable ...
3
votes
0answers
162 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 ...
4
votes
1answer
308 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
236 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
794 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 ...
7
votes
0answers
384 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 ...
6
votes
1answer
458 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) ...
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
781 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
452 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 ...
6
votes
1answer
684 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
837 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" ...
10
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 ...
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 ...
8
votes
1answer
500 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 ...