# Questions tagged [d-modules]

Modules over rings of differential operators.

196
questions

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### 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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**0**answers

55 views

### 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 ...

**1**

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**1**answer

58 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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81 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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77 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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93 views

### 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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186 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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49 views

### Parameterization of (homogeneous) twisted differential operators

Let $X$ be a smooth algebraic variety over a field $k = \bar{k}$ of characteristic $0$. It's well known that twisted sheaves of differential operators are parameterized by $H^1(X,\mathcal{Z}^1)$, ...

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90 views

### 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 $T^{*}X$ ...

**11**

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394 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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40 views

### 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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76 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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137 views

### 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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185 views

### 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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264 views

### 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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**1**answer

161 views

### 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{$\...

**6**

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**1**answer

444 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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59 views

### 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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79 views

### 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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**1**answer

89 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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170 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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149 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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53 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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153 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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61 views

### 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 $D$-...

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85 views

### Construction of non-split extension of simple modules of Lie algebras using linear differential operators

Consider the natural action of $W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}_2$-tuple ($\left[...

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**2**answers

369 views

### Explicit Riemann Hilbert correspondence

For simplicity, we assume that $X=\mathbb P_{\mathbb C}^1-\{s_1, s_2, \dots, s_k\}$ and $\infty \in X$.
Consider the trivial bundle $E=\mathcal O_X^r$ with the connection $\nabla$ induced by a ...

**3**

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**1**answer

145 views

### Reference request - existence of formal solutions for integrable connections

Let $K$ be a field of characteristic $0$, let $A = K[[t_1, \ldots, t_n]]$ be a power series ring over $K$, and let $V$ be a free $A$-module. Let $\nabla \colon V \rightarrow V \otimes_A \Omega^1_{A/K} ...

**5**

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283 views

### Equivalence of categories of $D$-modules on a singular $X$

Is $D^b(Mod_{qc}(D_X)) \to D^b_{qc}(D_X)$ an equivalence of categories for singular $X$? Where $Mod_{qc}(D_X)$ is the category of quasi-coherent modules on $X$ and $D^b_{qc}(D_X)$ is the full ...

**5**

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289 views

### Confusion on summand of Hochschild homology of D-modules

I've encountered the following confusion.
Start with the category of perfect D-modules on $\mathbb{A} ^{1}$, which I'll denote $D$. We have the object $\mathcal{O}$, which is exceptional in the sense ...

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**1**answer

143 views

### Riemann Hilbert Correspondence with fixed stractification

Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category ...

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40 views

### Holonomic D-module is locally artinian, what about the other implication?

Let $X$ be a complex manifold, and $M$ a coherent $\mathcal{D}_X$-module. Suppose that for all $x\in X$, the stalk $M_x$ is an artinian $\mathcal{D}_{X,x}$-module. Does it follow that $M$ is holonomic?...

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228 views

### Bernstein-Sato polynomial

Let $f$ be a polynomial. It is well-known that there exits a polynomial $b_f(s)$, such that $P\cdot f^{s+1}=b_f(s)f^s$ for some differential operator $P$. The polynomial $b_f(s)$ has been studied very ...

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301 views

### Remark 12.8.8 in Arinkin--Gaitsgory

I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully ...

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**1**answer

147 views

### Interesting (non) examples of singular support

I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...

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**1**answer

131 views

### Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$

Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked.
I know that there ...

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**1**answer

122 views

### Regular integrable connections and regular holonomic modules

I am trying to prove the following $\mathrm{Conn}^{\mathrm{reg}}(X) = \mathrm{Conn}(X) \cap \mathrm{Mod}_{rh}(\mathcal{D}_X)$.
Here an integrable connection on a smooth algebraic variety $X$ is a $\...

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131 views

### Functoriality and proofs

Recently, I began to study $\mathcal{D}$-modules and one of the major problems I encountered is that many natural transformations on the (derived) category of $\mathcal{D}$-modules are defined as the ...

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116 views

### Regular holonomic $\mathcal{D}$-modules

An holonomic $\mathcal{D}_X$-module on a smooth algebraic variety over $\mathbb{C}$ is called regular if all its composition factors are minimal extensions of the form $L(Y,N)$, where $N$ is a regular ...

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**1**answer

144 views

### Additivity of characteristic cycle of holonomic D-module

Let $\mathcal{M}$ be a holonomic D-module on a complex analytic (or alternatively, algebraic) manifold $X$. One can attach to it (using a good filtration) a characteristic cycle $Ch(\mathcal{M})$ ...

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152 views

### Is direct image of simple $D$-module is also simple?

(I have asked the question
The commutativity of minimal extension $\cdots$ and I simplify this question to the next simple question:)
Let $X$ be a rational variety over $\mathbb{C}$, $\phi : \hat{X} \...

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**1**answer

76 views

### The commutativity of minimal extension and direct image by blowing-down

Let $X$ be a sooth algebraic variety over $\mathbb{C}$.
Let us assume that there exists the commutative diagram
$\require{AMScd}$
\begin{CD}
U @>{i}>> \hat{X}\\
@| @VV{\phi}V\\
U @>{j}>&...

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349 views

### Kozsul resolution of $\mathcal{O}_X$

Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free ...

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174 views

### Is there a Lie group Fourier transform for D-modules?

Given a D-module $\mathcal{M}$ on $\mathbb{C}^n$, its Fourier transform $\widehat{\mathcal{M}}$ is equal to $\mathcal{M}$ as a set, but its module structure over $\mathbb{C}[x_1,...,x_n,\partial_1,...,...

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60 views

### Canonical map in the direct image of $\mathscr{D}_X$

Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...

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100 views

### Isomorphism classes of rings of differential operators

Let $X$ be your "favourite" kind of space, and let $\mathcal{D}_X$ be the (sheaf of) ring(s) of differential operators on $X$. What does the ring $\mathcal{D}_X$ tell us about $X$?
I know this might ...

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**1**answer

344 views

### Characteristic variety of a D-module

Let $X$ be an algebraic variety, $M \in Mod(\mathcal{D}_X)$. I am studying the characteristic variety associated to this module, and I am trying to understand why all the different definitions ...

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217 views

### Differential operators and quasi-finite morphisms

Let $ X, Y $ be smooth affine varieties over $ \mathbb C $. Let $ T : X \rightarrow Y $ be a dominant quasi-finite morphism and let $ T^\# : \mathbb C[Y] \rightarrow \mathbb C[X] $ be the resulting ...

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163 views

### A canonical isomorphism in derived categories of D-modules

I am learning D-modules recently, and my question might be technical. It arises from Lemma 2.6.13 in Hotta-Takeuchi-Tanisaki's book, which states that there exists a canonical isomorphism
$$
R\mathcal ...

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**1**answer

249 views

### Derived category of $\mathcal{D}_X$ modules

Define $D^{\text{#}}_{\bullet}(Mod(\mathcal{D}_X))$ to be the full subcategory of the derived category $D^{\text{#}}(Mod(\mathcal{D}_X))$ of complexes of $\mathcal{D}_X$-modules whose cohomology ...