# Questions tagged [d-modules]

Modules over rings of differential operators.

275
questions

3
votes

0
answers

171
views

### Computing pushforwards and pullbacks of D-modules

Let $X$ be a smooth algebraic variety (over some field of char 0), $Z$ a smooth closed subvariety of codimension 1, $i : Z \hookrightarrow X$ the inclusion, and $j : U \hookrightarrow X$ the ...

3
votes

0
answers

75
views

### Length of analytic holonomic D-modules

Fix a smooth complex algebraic variety $X$, and $\mathcal D_X$ its sheaf of differential operators.
If a $\mathcal D_X$-module $\mathcal M$ is holonomic, it has finite length. The usual proof uses the ...

5
votes

1
answer

156
views

### Quotient of the algebra of differential operators by an ideal generated by functions

Say $X$ is a smooth algebraic variety, and write $\mathcal D_X$ for the ring of differential operators on $X$ with the order filtration $F_{\bullet} \mathcal D_X$.
My problem is the following : I have ...

6
votes

0
answers

195
views

### Fourier transform for perverse sheaves

I am interested in studying the Fourier transform for perverse sheaves on "nice" spaces, say affine complex space stratified by the action of an algebraic group into finitely many orbits.
In ...

3
votes

0
answers

115
views

### Analytic analogue of implicit functions for differential operators

Let $p\colon \mathbb{R}^2 \to \mathbb{R}$ be a polynomial with a non-vanishing gradient at $p^{-1}(0)$. Then, the implicit function theorem says that $S = \{(x,y) \in \mathbb{R}^2 \mid p(x,y) = 0\}$ ...

5
votes

0
answers

174
views

### What advantages do perverse sheaves provide over D-modules? (or vice versa)

My question is as in the title: What advantages do perverse sheaves provide over D-modules? (or vice versa)
As a specific example: could something like the modular generalized Springer correspondence ...

4
votes

0
answers

120
views

### Introduction to the theory of $D$-modules and the role of the characteristic cycle

I am seeking recommendations for a concise introduction to the theory of $D$-modules suitable for an algebraic geometer. Specifically, I am interested in understanding:
The role of the characteristic ...

1
vote

0
answers

54
views

### Quantisation of shifted cotangent bundles

The cotangent bundle $T^*X$ of a smooth space $X$ quantises (e.g. in the deformation quantisation sense) to the sheaf $D_X$ of differential operators on $X$.
What is the analogous quantisation of the ...

1
vote

0
answers

151
views

### How does the cohomology theory on de Rham (pre)stack compute de Rham cohomology?

Recently when I'm reading PTVV, Shifted Symplectic Structures, in Sec. 2.1 Mapping stacks, the authors use the identification $H^*(X,E)\simeq H^{*}_{dR}(Y/k,\mathcal{E})$ to show $X=Y_{dR}$ admits an ...

11
votes

1
answer

912
views

### Reference for a statement from Gaitsgory's thesis

In his PhD thesis, Gaitsgory (in his "remark 6") makes the following claim:
Consider two complexes of holonomic $D$-modules with regular singularities on a variety $X.$ Suppose that at each ...

3
votes

1
answer

361
views

### Riemann-Hilbert problem via quiver description

The moduli space of Fuchsian systems over $\mathbb{P}^1$ with prescribed adjoint orbits conditions at poles a.k.a. additive Deligne-Simpson problem can be presented under purely quiver description.The ...

4
votes

1
answer

226
views

### Are perverse sheaves representations of some topological invariant?

The well-known correspondence between vector bundles with flat connection on a smooth complex algebraic variety $X$ and complex representations of $\pi_1(X^{an})$, the fundamental group of the ...

3
votes

1
answer

472
views

### F-crystals from crystalline cohomology

In Section 7 of Katz' paper:
https://web.math.princeton.edu/~nmk/old/travdwork.pdf
He asserts "Crystalline cohomology tells us that for each integer $i \geq 0$, the de Rham cohomology $H^i=Rf_*(\...

2
votes

0
answers

116
views

### Lie Algebra representations outside of generalized central characters

For a simple Lie algebra $\mathfrak{g}$, we can view its category of representations as fibered over $\operatorname{Spec}Z(\mathfrak{g})$ (a representation will lie over a point if the center's action ...

2
votes

0
answers

105
views

### Applications of the Riemann-Hilbert Correspondence

I am aware of the (first) proof of the Kazhdan–Lusztig conjectures using the Riemann-Hilbert Correspondence. Are there any other interesting applications of the RH correspondence?

2
votes

0
answers

309
views

### Modern treatment of $q$-differential operators/$\mathcal{D}_q$ modules?

The basic idea of $q$-differential operators: replace
$$\partial\cdot x^n\ =\ nx^{n-1} \hspace{10mm}\rightsquigarrow\hspace{10mm} \partial\cdot x^n \ =\ [n]_q x^{n-1} $$
where $[n]_q=(q^{n}-1)/(q-1)$ ...

3
votes

0
answers

232
views

### What is a twisted D-module?

Let $X/\mathbb{C}$ be an abelian variety, $Y$ be the dual abelian variety, and $P$ be the Poincaré bundle on $X\times Y$. On p.207, Correction to “Sheaves with connection on abelian varieties” (by M. ...

2
votes

0
answers

125
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.
...

9
votes

0
answers

455
views

### What's a holonomic D-module from the point of view of de Rham spaces?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We can consider its de Rham space $X_\text{dR}$ as the sheaf on $(\textsf{Sch}/\mathbb{C})_\text{ét}$ defined by $X_\text{dR}(S):= X(S_\text{...

4
votes

0
answers

188
views

### D-modules generated by derivatives of Delta function

We consider D-modules over the affine line $\mathbb{A}_{\mathbb{C}}^{1}$, i.e. modules of Weyl algebra $A_1(\mathbb{C})=\mathbb{C}[x,\partial]$. There is a set of microfunctions (ref. A Primer of ...

4
votes

0
answers

251
views

### Confusion about definition of crystals

In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...

8
votes

1
answer

288
views

### What are twisted Verma modules? Basic properties?

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...

13
votes

2
answers

2k
views

### Can we use Mann's six-functor formalism with D-modules?

In a recent course in Bonn, P. Scholze explains a formalization of a six-functor formalism due to L. Mann. In this axiomatization, three of the functors $f_!,f^*,\otimes$ are "constructed" (...

2
votes

1
answer

225
views

### D modules over nodal curves

Is there a simple description for $D$ modules over $\text{Spec}\left(k\left[x,y \right] / \left(xy\right) \right)$ (say k is algebraically closed of characteristic 0)? Is there a description of the $D$...

3
votes

0
answers

110
views

### Local cohomology with coefficients in ideals of parameters

I'm not an expert in local cohomology, but the following problems have come up in my work, and I'd like to get a sense of where things stand.
Let $\mathbb{A}^n=\operatorname{Spec} \mathbb{C}[x_1, \...

4
votes

0
answers

102
views

### Confusion about D-affineness and jet sheaves on projective line

I have encountered what feels like a very basic confusion relating to the Beilinson Bersnstein localisation theorem.
This theorem in particular states that if $F$ is a $D_{\mathbb{P}^{1}}$ module, ...

2
votes

0
answers

166
views

### Smooth pullback of holonomic D-modules is fully faithful

Let $X$ and $Y$ be (not necessarily smooth) algebraic varieties over an algebraically closed field of characteristic $0$, and suppose we have a smooth surjective map $f: X \to Y$ of relative dimension ...

2
votes

0
answers

100
views

### Generalizations of elliptic chain complexes

I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....

1
vote

0
answers

93
views

### Good D-module is induced by coherent O-module?

Let $X$ be a compact complex manifold. In Definition 4.24, p.78 of D-modules and Microlocal Calculus (by Kashiwara), a coherent $D_X$-module $F$ is called good if there is a directed family $\{G_i\}...

2
votes

1
answer

213
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^\...

3
votes

1
answer

271
views

### Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?)

Let $f:X\to S$ be a morphism between algebraic varieties which are smooth over a field of characteristic zero. We define the (derived) direct image functor $f_+:\mathsf{D}^b(\mathcal{D}_X)\to \mathsf{...

6
votes

0
answers

308
views

### D-modules on singular varieties; forgetful functors, and t-structures

Let $Z$ be a singular variety over the complex numbers with a closed embedding $i: Z \hookrightarrow X$ into a smooth variety $X$. One can define the derived category $\mathcal{D}(Z)$ of D-modules on $...

4
votes

0
answers

96
views

### Holonomic distributions in the analytic setting

We are interested in a reference to the notion of Holonomic distributions in the real analytic setting. Namely, distributions that generate a Holonomic D-module under the action of the algebra of ...

1
vote

1
answer

272
views

### Every rank 1 local system $L$ satisfying $m^*L=L\boxtimes L$ comes from the Lang torsor? The same holds for D-modules?

Let $G$ be a commutative connected algebraic group over a field $k$, with group operation $m:G\times G\to G$. If $k=\mathbb{F}_q$, we may use a character $\varphi:G(k)\to\overline{\mathbb{Q}}_\ell^\...

5
votes

0
answers

151
views

### Concrete computation of pullback of the $D$-module $\mathbb{C}[t,t^{-1}]$

I have some problems in calculating some example explicitly. Consider
$$ \{0\} \overset{i}{\rightarrow} \mathbb{C} \overset{j}{\leftarrow} \mathbb{C}^*.$$
Then $Rj_+\mathbb{C}[t,t^{-1}] = \mathbb{C}[t,...

3
votes

1
answer

158
views

### Explicit computation of D-modules pullback

Consider the $D_{\mathbb{A}^1}$-module $M:=D_{\mathbb{A}^1}/(x)$, and the map $f:z\mapsto z^k$. I want to know $f^*(M)$. I believe it only has a single non-zero cohomology, namely in degree $0$, which ...

2
votes

1
answer

182
views

### Confusion about Wakimoto's chiral differential operators on $\mathbf{P}^1$

It is a classical result of Wakimoto that the sheaf of chiral differential operators $D_{ch}$ on $\mathbf{P}^1$ has global sections
$$D_{ch}(\mathbf{P}^1)\ \simeq\ L_{-2}(\mathfrak{sl}_2)$$
the simple ...

5
votes

1
answer

263
views

### Two identities involving Ext functors in the context of D-modules

I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $X$ be a complex affine algebraic variety ...

5
votes

1
answer

269
views

### Holonomic = annihilated by some differential operator

Let $X$ be a smooth variety over a characteristic zero field $k$. It seems to be well-known that
"A coherent $\mathcal{D}_X$-module is holonomic if and only if 'every element is annihilated by a ...

4
votes

0
answers

142
views

### D-module theoretic Chern characters and an index-type theorem

Let $X$ be a smooth projective variety over $\mathbf{C}$. Consider the category $\mathbf{D}_{X}^{\text{perf}}$ of perfect complexes of left $D$-modules on $X$. It is well known that there is an ...

4
votes

1
answer

327
views

### Compatibility between the functors of $\mathcal{O}_X$-modules and $\mathcal{D}_X$-modules

Let $f:X\to Y$ be a morphism between smooth algebraic varieties over $\mathbb{C}$.
We have natural functors $f^!:\mathsf{D}_{\text{qc}}(\mathcal{D}_Y)\to \mathsf{D}_{\text{qc}}(\mathcal{D}_X)$, $f_*:\...

3
votes

1
answer

342
views

### Koszul complex of a $\mathcal{D}$-module

I am trying to understand the construction of push forward in the derived category of $\mathcal{D}$-modules using the Koszul complex. My question will be about the latter notion.
Let $X$ be a smooth ...

7
votes

0
answers

203
views

### A reference for Bernstein's approach to KL conjectures

The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy.
Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...

2
votes

1
answer

303
views

### Open/closed embeddings and the de Rham space

Let $U\to X$ be an open immersion of schemes and denote by $D$ the (say reduced) complement. Then by applying the de Rham functor, we get morphisms
$$U_{dR}\to X_{dR}\leftarrow D_{dR}$$
of the ...

3
votes

0
answers

278
views

### An algebraic proof: A line bundle on a curve with a connection must be of degree 0

Let me state it using the language of $D$-modules. Let $X$ be a smooth projective curve and $\mathcal{L}$ a line bundle on it. Assume that $\mathcal{L}$ has a left action of $\mathcal{D}_X$. Then show ...

3
votes

1
answer

247
views

### do hyperfunction solutions always exist?

I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...

4
votes

1
answer

579
views

### Six functor formalism for quasi-coherent $D$-modules

Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}...

8
votes

1
answer

543
views

### D-modules as ind-coherent sheaves over positive characteristics?

There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I....

6
votes

0
answers

426
views

### Definition of the tensor product of $D$-modules

Let $\mathsf{D}^b_h(X)$ be the full subcategory of $\mathsf{D}^b(D_X\textsf{-Mod})$ (the bounded derived category of left $D_X$-modules) consisting of the complexes with holonomic cohomology. The ...

6
votes

0
answers

266
views

### The geometric "hands-on" vs. algebraic approach to nearby cycles

Feel free to skip to the question below; the following is just context and discussion:
An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (...