# Questions tagged [d-modules]

Modules over rings of differential operators.

221
questions

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### On the use of the fundamental exact sequence of K\"ahler differentials in a paper of Lyubeznik

Let $k$ be a field, $R := k[x_1, \cdots , x_n]$ the polynomial ring in $n$ indeterminates over $k$ and $f$ a nonzero element of $R$. The following paper of Lyubeznik which I have been recently reading,...

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66 views

### Socle of a quotient of the ring of differential operators of a polynomial ring

I have been reading the following paper:
https://www.sciencedirect.com/science/article/pii/S002240491000263X
Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...

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179 views

### Cohomological dimension for stacks

If $X$ is a scheme (maybe with conditions), I'm pretty sure that the ($\ell$-adic/de Rham) rational cohomology $H^*(X,\mathcal{F}$) of an $\ell$-adic sheaf/holonomic $D$-module $\mathcal{F}$ vanishes ...

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127 views

### Geometric interpretation of $\mathbb{C}^{\times}$-gerbes

Let $X$ be a (nice enough, e.g. smooth etc.) variety over the complex numbers, and let $\mathcal{G}$ be a gerbe on $X$. Then $\mathcal{G}$ is classified by a cohomology class in $\alpha \in H^2(X, \...

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239 views

### Symmetric powers of curves and completion along the diagonal

Given a smooth curve $C$, denote by $\text{Sym}^d(C)$ its $d$-th symmetric power. Let $\Delta$ be the diagonal subvariety which is defined as the codimension $1$ subvariety that at least two of the ...

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### $K$-theory of $D$-modules

I have to admit I don't know much about topics appearing in this question, I just see very rough connections between these objects:
According to this page 23, a different $t$-structure on $D^b(\text{...

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**1**answer

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### Is analytification of regular holonomic D modules a fully faithful functor?

It seems to me that by the algebraic Riemann Hilbert functor(which factors through the analytification map) and also the analytic Riemann Hilbert functor that the (derived) category of (algebraic) ...

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160 views

### Is the abelian category of pure Hodge modules semi-simple?

I am a beginner in the subject, and at the moment I am trying to understand basic properties of the main objects of the M. Saito's theory of the mixed Hodge modules in general.The question in the ...

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154 views

### Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$?

When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral
$$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$
where $P$ is a linear ...

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93 views

### Are there six functors for twisted D modules?

Is there a notion of holonomic D module which admits the six functor formalism in the world of twisted D modules?
Recall that twisted D modules on $X$ are well-defined for any $T$ torsor $\tilde{X}\...

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87 views

### Left $D$-modules for singular varieties?

I asked the same question on MSE but got no answer, I hope it's ok to ask it here.
In these notes one can read page $5$ "it may seem surprising, but one cannot define left $D$-modules for non-...

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126 views

### BRST cohomology and vanishing cycles

Consider the $\mathbb{C}$-variety $\mathbb{A}^{1}$, equipped with the potential (ie global function) $P:=\frac{z^{n+1}}{n+1}$. We can form the twisted de Rham complex $H_{dR}(\mathbb{A}^{1},P)$ which ...

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54 views

### Reference request: direct image of equivariant $\mathcal D$-modules

I'm hoping to find a detailed proof for the following well-known fact:
Given respective algebraic group actions $G_1, G_2$ on smooth varieties $X_1, X_2$ over $\mathbb C$ with an algebraic group ...

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78 views

### Computing $\mathcal D$-module direct image along group action map

Say everything is over $\mathbb C$, and I have an action $act: N \times X \to X$ of an affine algebraic group $N$ on a smooth variety, say with finitely many orbits. I'm trying to compute the $\...

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51 views

### Tensor product of modules over twisted differential operators

Let $R$ be an algebra over complex numbers. Let $N$ be a module over $R$. We can define the algebra $D(N)$ of differential operators $N \rightarrow N$ using Grothendieck’s approach as follows: we ...

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235 views

### Absolute purity for intersection cohomology

If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then
$$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$
where $(\star)$ is true when $i$ is in addition regular.
Here ...

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**1**answer

165 views

### Natural map from vector fields to cotangent variety

Let $X$ be a smooth variety, and let $\operatorname{Vec} (X)$ denote the $\mathcal{O}_X$-module of vector fields on $X$. It is stated in several books on D-modules, for example here in Corollary 6.6, ...

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241 views

### $D(\mathcal{O}(n))$ via generators and relations

Let $V$ be a complex vector space. Consider the algebra $D(\mathbb{P}(V),\mathcal{O}(n)))$ of global differential operators from line bundle $\mathcal{O}(n)$ to itself, here $n \in \mathbb{Z}_{\...

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### Quasi-unipotent monodromy for variation of Landau-Ginzburg cohomology

A pair, $(X,f)$, consisting of a smooth variety and a global function $f:X\rightarrow\mathbb{A}^{1}$ is called a Landau-Ginzburg model, LG-model for short. The LG-cohomology of the pair, dentoed $H(X,...

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53 views

### Naive pushforward of D-modules and Gauss--Manin connection

Suppose that $f\colon X\to Y$ is a morphism of smooth quasi-projective varieties over a field of characteristic $0$. We then have a naive pullback functor $f^\circ:=\mathcal D_{X\to Y}\otimes_{f^{-1}\...

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74 views

### $G$ invariant of $D$-module direct image of the structure sheaf under quotient map

Let $G$ be a finite group acting on $V$, a complex affine variety. Suppose $\pi:V\to V/G$ is the quotient map. $V/G$ is most likely singular, consider a map $i:V/G\hookrightarrow Y$ where $Y$ is ...

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195 views

### When is cohomology of a finitely presented dg-algebra computable?

Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...

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82 views

### Is this definition of a Fuchsian operator correct?

In Bjork, Analytic D-modules and applications, the following definition of a Fuchsian operator is given:
Here, I believe, $D(0)=\mathcal{O}$, the zeroth filtered piece of the ring of germs of ...

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**1**answer

438 views

### What's the relationship between a $E_2$-Hochschild Cohomology module and a D-module?

Let's say for simplicity $A$ is a smooth algebra over a field $k$ ($A$ and $k$ are discrete commutative rings but from now on we are fully derived), and we will consider the $E_2$ algebra $HH^{\bullet}...

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116 views

### On regular holonomic $D$-modules on the disk

Denote by $\Delta\subset \mathbb{C}$ a small neighborhood of $0$. Let $M$ be a regular holonomic $D_\Delta$ module. In Bjork, "Analytic D-modules and applications", section 5.2, it is proven ...

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120 views

### Relations between $\mathcal D$-modules and Exterior Differential Systems?

As a followup of this answer, I wonder relations between $\mathcal D$-modules and Exterior Differential Systems, both of which could be used to describe systems of PDEs. If I understand correctly, $\...

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426 views

### Examples of Stokes data

I'm trying to learn Stokes data but can't find an example to get my teeth into it.
Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence
$$\text{regular ...

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### Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it.
Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...

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113 views

### Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface

I originally posted the question on math.stackexhange, but there doesn't seem to be an answer. I apalogize in advance for cross posting.
Let $E\rightarrow X$ be a holomorphic vector bundle over a ...

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97 views

### Compare bounded (unbounded) derived categories of D-modules

I am reading [HTT, D-modules, Perverse Sheaf and Representation Theory]. In [HTT, 1.5.7, page 32], it is claimed that an equivalence of categories between bounded derived category with quasi-coherent ...

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81 views

### The canonical morphism on the nearby cycle of a D-module

I am looking for some references of the following fact:
For a regular holonomic D-module $M$, let $V_\bullet M$ be the V-filtration wrt a smooth hypersurface $t=0$. Then the complex $\partial_t: gr^...

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### Operations on perverse sheaves on disk

The category of perverse sheaves on the disk is isomorphic to the category of diagrams
$$E\substack{\substack{c\\\to}\\\substack{v\\\leftarrow}}F$$
With $E,V$ finite dimensional vector spaces, and ...

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280 views

### On Grothendieck's abstract definition of differential operators

I have heard that there is the following abstract definition due to Grothendieck of differential operators on a module $M$ over a commutative associative unital algebra $A$ over a field of ...

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### singular support in the singular case

For any constructible sheaf (or D-module) $\mathcal{F}$ over a smooth variety $X$ over $\mathbb{C}$, there is a notion of singular support $SS(\mathcal{F})$ that lives in the cotangent bundle $T^{*}X$ ...

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602 views

### Factorization and vertex algebra cohomology

A chiral algebra on a smooth curve $X$, in the sense of Beilinson-Drinfeld, is a right $D_{X}$-module with a chiral bracket, which is a map $\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta_{*...

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### Is the characteristic cycle map for perverse sheaves injective?

Let $X$ be a smooth irreducible complex variety. Is the characteristic cycle map from the Grothendieck group of perverse sheaves (with complex coefficients) on $X$ to the free abelian group generated ...

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81 views

### Embedding representation of fundamental group in functions on universal cover

Let $X$ be a complex manifold, $Y\rightarrow X$ the universal cover, and $\Gamma$ be the Galois group of this cover, ie the fundamental group of $X$.
Finite dimensional $\Gamma$-representations ...

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### Solutions to holonomic $D$-modules: when are they square-integrable?

I want to apply the theory of $D$-modules to solve operator equations of several variables in the Bargmann space
$$\mathcal H :=\bigg\{\psi \in \mathcal O^\text{an}_{\mathbb{C}^n}\,\,\bigg|\,\,\,\...

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### Can we reconstruct a regular variety from its $D$-$\mathrm{mod}$?

The famous Gabriel-Rosenberg reconstruction theorem states that the pre-additive category of quasi-coherent sheaves on a quasi-separated scheme contains enough information to recover the scheme.
One ...

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### 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, ...

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### Testing ideal membership in the Weyl algebra: a simple example

In Example 1.1.4 of the book Grobner Deformations of Hypergeometric Differential Equations, it is stated without proof that
$$\partial^2 \in D\cdot \langle x\partial^4, x^3\partial^2 \rangle \tag{$\...

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464 views

### Category of $\mathcal{D}$-modules on a singular variety

Take $X\to V$ a closed embedding, where $X$ is not necessarily smooth, $V$ is affine and smooth. Define the category $\mathcal{C}$ of $\mathcal{D}$ modules on $X$ to be the full subcategory of $\...

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### Roots of b-function and vanishing of D-modules

Let $f$ be a polynomial in $n$ variables with complex coefficient. The $b$-function (or Bernstein-Sato polynomial) is the minimal nonzero monic polynomial $b_f(s)$, such that there exists differential ...

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### What is the precise relationship between projective duality and the Radon transform?

The Radon transform I am referring to is the one appearing in Brylinski's paper on projective duality, using the incidence correspondence over projective space and its dual projective space, $R p_{2*} ...

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105 views

### Pullback of homogeneous twisted differential operators

Let $X,Y$ be smooth complex varieties and let $G$ be an smooth affine algebraic group acting on $X$ and $Y$ such that $X,Y$ are $G$-homogeneous spaces (the $G$ action is transitive). We also let $f:Y \...

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206 views

### Proof of Kashiwara's constructibility theorem for algebraic D-modules

I am trying to understand the proof of Kashiwara's constructibility theorem for algebraic D-modules, following either the book "D-modules, Perverse sheaves, and representation theory" by Hotta, ...

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151 views

### How to define a “truncated solution complex” $RHom_{D_{X,x}}(M_x,\mathcal{O}_{X,x}/\mathfrak{m}_x^k)$?

Let $M$ be a regular holonomic $D_X$ module on a smooth complex variety $X$. The comparision theorem says that
$$RHom_{D_X}(M,\mathcal{O}_X)_x\cong RHom_{D_X,x}(M_x,\hat{\mathcal{O}}_{X,x}).$$
Now ...

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58 views

### Endomorphism of holonomic $D$-module has minimal polynomial

Let $M$ be a holonomic $D_n$ module, where $D_n=\mathbb{C}[x_1,\partial_1,\dots,x_n,\partial_n]$, so we work in affine space. What is the easiest way to prove that any endomorphism of $M$ has a ...

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164 views

### For a holonomic $D_X$-module $M$, can $\operatorname{gr}M$ have embedded primes?

Let $M$ be a holonomic $D_X$-module. This means that the minimal primes in $\sqrt{\operatorname{Ann}(\operatorname{gr}M)}$ are $n=\dim X$ dimensional, for some (and any) good filtration on $M$. But ...

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### Is there some nice “Nakayama-like” method to test whether a holonomic D module is 0?

A holonomic $D$-module is either the zero module, or one for which the characteristic variety is of the smallest possible dimension. Is there some useful trick to determine whether a holonomic $D$-...