The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
122 views

Characteristic Varieties and Associated Varieties

Two notions that occur often in representation theory seem to be that of a "characteristic variety" and that of an "associated variety". The former term seems exclusive to D-module theory while the ...
2
votes
1answer
135 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
93 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 ...
1
vote
0answers
80 views

$D^{\infty}$ modules on analytic spaces

In Mebkhout's paper on Local Cohomology of Analytic Spaces, the following theorem is stated: Let $X$ be a complex smooth manifold and $Y$ is an analytic subspace of $X$. Then ...
11
votes
0answers
241 views

Epsilon factors - a la Beilinson - What is it?

I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or ...
11
votes
2answers
463 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
4
votes
1answer
195 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
95 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 ...
1
vote
1answer
53 views

Projective volume form

Upon reading K. Costello's paper on Witten genus, I wonder when, on a smooth (quasi-)projective variety $X$, the canonical bundle $\omega_X$ admits a left $D$-module structure (other than the ...
11
votes
2answers
479 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 ...
3
votes
1answer
388 views

The Weyl algebra modules which are also rings

Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...
1
vote
1answer
161 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 $k = \mathbb{C}$.) ...
3
votes
0answers
46 views

Convolution of DQ-Modules

On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push ...
9
votes
1answer
254 views

Bernstein-Sato polynomial (one variable)

Let $R = \mathbb{C}[x_1,...,x_n]$, $p \in R$. There exists a monic (of lowest degree) $b_p(x) \in \mathbb{C}[s]$ and a differential operator $D(s)$ such that $$b_p(s) p^s = D(x)p^{s+1}.$$ The ...
1
vote
0answers
119 views

The associated graded of a mixed Hodge module

Unfortunately this question will be a bit vague, since the question revolves around a memory of something I may have heard in a talk (long time ago). Let $X$ be a smooth complex variety. Let $MHM(X)$ ...
1
vote
0answers
83 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} ...
3
votes
0answers
96 views

Is there an analogue of distributions in characteristic p?

Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...
4
votes
0answers
187 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
0answers
118 views

A nice rigid analytic model for local systems over an elliptic curve?

For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts ...
2
votes
2answers
298 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. ...
1
vote
0answers
228 views

D-modules as quantization of modules on cotangent bundle

If we have filtration on D-module compatible with some good filtration on differential operators sheaf then adjoint graded module is module over functions on cotangent bundle. So morally D-module is ...
2
votes
1answer
244 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
235 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
445 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
345 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
169 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 ...
9
votes
0answers
287 views

Duality in category O vs. Duality of D-modules

Hello, I omit in the following all the words "derived, twisted, holonomic, finitely-generated...". We have the Bernstein-Beilinson equivalence between the category of $N$-equivariant $D$-modules on ...
4
votes
1answer
291 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 ...
3
votes
1answer
358 views

Polarizable variations of (mixed) Hodge structures

I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be ...
1
vote
0answers
359 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
617 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
214 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 ...
5
votes
1answer
385 views

What kind of algebra has geometric realization as in “Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups”

In Chriss and Ginzburg's book "Representation Theory and Complex Geometry" as well as the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups", the group algebra ...
1
vote
0answers
144 views

Characteristic variety of a D-module along smooth pullback

All varieties are over the complex numbers. Given a smooth variety $X$, write $T^* X$ for its cotangent bundle. For a morphism of smooth varieties $f: X \to Y$ write $f_{\pi}: T^*Y \times_Y X \to ...
6
votes
0answers
215 views

Non-characteristic maps (ala D-modules)

I am trying to understand a `well known' fact (see Kashiwara'sIntroduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is ...
11
votes
1answer
615 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
223 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 ...
3
votes
0answers
156 views

Induced D-modules

According to the usual definition, an induced D-module on a complex manifold $X$ is a right D-module of the form $L \otimes_{\mathscr{O}_X} \mathscr{D}_X$, for $L$ a coherent sheaf of ...
9
votes
2answers
715 views

Does the Riemann-Hilbert Correspondence work at the DG level?

let $X$ denote a smooth complex algebraic variety. Let $D_{rh}(X)$ denote the category of regular holonomic $D$-modules on $X$ and $D_{rh}^b(D(X))$ denote the bounded derived category of $D$-modules ...
5
votes
0answers
167 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, ...
3
votes
2answers
559 views

How is this observation related to Koszul duality?

Let $X$ be a smooth variety, $\mathcal D$ the sheaf of algebraic differentail operators, $\Omega$ the algebraic deRham complex and $\mathcal M$ a quasi coherent $\mathcal O_X$-module. Now there is a ...
4
votes
1answer
428 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} ...
1
vote
1answer
315 views

existence of global good filtration for D-modules?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$ (or a field of characteristic zero). We have $D_X$ the sheaf of differential operators on $X$, which is a coherent sheaf of rings, and it ...
6
votes
1answer
406 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
489 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
874 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 ...
3
votes
0answers
138 views

Characteristic cycle of irreducible holonomic modules

Let $i:A\rightarrow X$ be an affine locally closed inclusion of smooth complex varieties. What can be said about the characteristic cycle of the minimal extension $i_{*!}\mathcal{O}_A$? How would one ...
12
votes
2answers
911 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
330 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 ...
0
votes
0answers
188 views

Does regularity of a D-module for an unusual filtration imply regularity for the usual one?

One definition of regular D-modules on affine space is that a D-module is regular if it has a filtration compatible with the order filtration on differential operators whose associated graded is ...