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Questions tagged [d-modules]

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Weyl algebra acting on $m$-tuple of polynomials

Let $\mathscr D=\mathbb R[x, \partial]$ be the Weyl algebra of one variable over the reals. Suppose $m$ is a positive integer, $b_1(x), \ldots, b_m(x)$ are linearly independent polynomials in $\mathbb ...
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to which intrinsic object corresponds connection's hypersurface

Given a complex manifold $M$ and an hypersurface $S$, and some connection on the line bundle associated to $S$, to which intrinsic object of $S$ corresponds the connection ? (more specifically, same ...
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suggest a course on differential operator and Weyl

Can someone suggest me a course in pdf or textbook about differential operator and Weyl algebra. let $ R$ a commutative algebra over a field $k$ let M, N two R-modules. we define the set of ...
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1answer
59 views

Transfer modules for closed embeddings

Given a map $f:X\to Y$ of smooth complex manifolds, the transfer module $\mathcal D_{X\to Y}$ is the left $\mathcal D_X$-module $\mathcal O_X\otimes_{f^{-1} \mathcal O_Y} f^{-1} \mathcal D_Y$. I'm ...
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1answer
37 views

Weyl algebra $\Bbb C[[x]][\partial]$ and division

Let $R = \Bbb C[[x]]$ the ring of formal power series and $A = R[\partial]$ the ring of differential operators with the relation $[\partial,x] = 1$. There is the following proposition in my book ("...
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38 views

Global sections of pullback of $G$-equivariant $D_Z$-modules

Let $G$ be a semi-simple complex algebraic group with lie algebra $\mathfrak{g}$. For a fix Borel subgroup $B$ let $X=G/B$ be the flag variety. Let $i_l,i_r:X \to X \times X=Z$ denote the inclusion of ...
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32 views

looking for a proof or a reference of classical result

Let $X$ be a complex manifold of dimension $n$, $\Omega_{X}$ the sheaf of top degree forms, $\mathcal{D}_{X}$ the sheaf of holomorphic differential operators of infinite order and $q$ the natural ...
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31 views

looking for flabby sheaf resolutions

I am looking for manipulable flabby resolution of the sheaf of top-degree forms (let say on a complex manifold $X$) which is not canonical, i.e. not the Godement resolution. Does it exist any ...
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Inverse image of meromorphic connections

Let $X$ be a complex manifold and $D$ a divisor on $X$. Let $f : Z \rightarrow X$ be a morphism of complex manifolds and assume that $f^{-1}D$ is a divisor on $Z$. A meromorphic connection on $X$ ...
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254 views

The relative de Rham complex

Here (at the bottom of page 13) it's stated that for a smooth map $f:X\to S$, the relative Spencer complex $$\Omega_{X/S}^\bullet \otimes_{f^{-1}\mathcal{O}_S} D_X$$ is a resolution of the transfer ...
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1answer
100 views

Are there holonomic $\mathcal{D}$-modules besides flat connections?

Question: are there interesting holonomic $\mathcal{D}$-modules on smooth variety $X$ except those coming from flat meromorphic connections? $$\text{}$$ Since $M$ is holonomic iff $\dim \text{ch}(M)...
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Transfer modules and Weyl algebra

Let $V$ be a $\mathbb{C}$-vectorial space of dimension $n$ and $V^*$ the complex dual space. I would like to understand the following isomorphism $$D_{V^* \leftarrow V \times V^*} \overset{L}\otimes_{...
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1answer
41 views

Inverse of the sheaf $\Omega_X$

Studying $D$-modules, I often see the notation $\Omega_X^{\otimes -1}$ where $X$ is a complex manifold and $\Omega_X$ are the top holomorphic forms on $X$. If I understand it well, this sheaf verifies ...
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94 views

The D module pushforward to a point is de Rham cohomology?

I have read and understood the definition of the pushforward of a $\mathcal{D}_X$-module $\mathcal{M}$ under a morphism of varieries $$f:X\longrightarrow Y$$ In particular, if $Y=\{ \star \}$ is a ...
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151 views

Beilinson-Bernstein localization, equivariant modules

I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a field of charateristic $0$ and $K$ a closed subgroup of $G$ ...
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45 views

Algebraic cohomology $R\Gamma_{[Z]}(\cdot)$ for locally closed analytic set $Z$

I really searched for this quite a while and didn't find an answer - I hope I didn't miss anything obvious. Let $X$ be a complex manifold ($\mathscr O_X$ the sheaf of holomorphic functions on $X$) ...
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88 views

“D-module'' or ”$D$-module''?

Disclaimer: This question is not transcendental at all, so go easy with me. Starting with a (for simplicity) commutative unital ring $R$, we define a $R$-module. Obviously, since the name of the ring ...
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Proving a certain $ \mathcal{D}$-module is holonomic

Studying a certain system of hypergeometric partial differential equations, I came across the proof that the $ \mathcal{D}$-module associated to it is holonomic. My issue it that I can't follow the ...
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1answer
90 views

A question regarding an application of the Seidenberg-Tarski Theorem

I'm reading J. N. Bernstein's Paper; The Analytic Continuation of Generalized Functions With Respect to a Parameter. You can find it here: http://www.math1.tau.ac.il/~bernstei/Publication_list/...
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1answer
106 views

How can I give an explicit presentation of the ring of differential operators over a smooth variety?

I'm trying to find a presentation of the ring of differential operators for a smooth affine and smooth projective variety. For example, consider the varieties \begin{align*} X = \textbf{Spec}\left( R =...
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1answer
367 views

Canonical connection on the Frobenius pull-back

If $X$ is a scheme over a scheme $S$ of characteristic $p>0$ and $F:X\to X^{(p)}$ is the relative Frobenius, it is known that there is a canonical connection on the Frobenius pull-back $F^*E$ of a ...
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60 views

Operator that kill the wave function

I have the following function in $x$. $\sum_{d=0}^{\infty} \frac{1}{\hbar^d}\frac{1}{d!}\left(\prod_{i=1}^{d-1}(1+i\hbar)^{m}\right)x^d$ I need a differential operator involving $(x,\frac{d}{dx},h,...
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1answer
71 views

Why does this identity hold for D-modules?

Consider $\mathbb{C}^n$ with coordinates $x_1,\ldots,x_n$ and let $\partial_1,\ldots,\partial_n$ be the corresponding vector fields. Then the canonical free rank 1 D-module should be $$ \mathbb{C}[x_1,...
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Bernstein polynomial of $x_1^2+\cdots+x_n^2$

According to S. C. Coutinho’s A Primer of Algebraic $D$-modules (see p. 95), the Bernstein polynomial of $p=x_1^2+x_2^2+\cdots+x_n^2\in K[x_1,\dots,x_n]$ (with $K$ an arbitrary field of characteristic ...
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164 views

Characteristic Variety of the Principal Symbol solves PDE system?

In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see, https://en.wikipedia.org/wiki/...
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1answer
76 views

When is a D-module holonomic?

Suppose that I have a small family of partial differential equations such that $y(x)$ must be a solution to $$ P_i(x,D) y(x)=0,\ \ \ \ \ \ i=1,...,m $$ for all $i=1,...,m$, where I am using multi-...
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1answer
575 views

Defining a sheaf of differential operators

I have two questions related to the coordinate-free definition of $\mathcal{D}$-modules provided in Ginzburg's notes (see pages 24-25): Exercise 2.1.13 says that Diff($M,M$) is almost-commutative, i....
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1answer
102 views

A question on Weyl algebra $A_1$

This question is from A Primer of Algebraic D-Modules by S.C. Coutinho, Ex 2.4.10. In the following $A_1$ means the Weyl algebra generated by $x$ and $d$. Let $f: A_1^2\to A_1$ be the map defined ...
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85 views

Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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1answer
52 views

Space of polynomial solutions of $P(f) = 0$ for $P \in A_1 (\mathbb{C})$ has finite dimension

I've been trying to work through Coutinho's Primer on Algebraic D-Modules and I'm having trouble with proving the following exercise: Let $P \in A_1 (\mathbb{C})$, the first Weyl Algebra over $\...
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1answer
282 views

holonomic D-modules

I am trying to develop an intuition about holonomic D-modules and find the literature formidable (I study physics). My question is, given a linear differential operator in n-variables, $x=(x_1,...,...
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1answer
237 views

A problem in elementary calculus

Let $P(x), Q(x)$ be two polynomials with real coefficients and set $F(x) = \frac{P(x)}{Q(x)}$. Consider a table which has the function $e^{\int_0^x F \, dx}$ on it. The table has the set of rules that ...
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1answer
231 views

Is it really unknown that every endomorphism of the Weyl algebra $A_1$ is an isomorphism?

Here $A_1 := K\{x\cdot-, \frac{d}{dx}\} \subset \operatorname{End}_K(K[x])$ for some characteristic-zero field $K$. I found this claim in Coutinho's "A Primer of Algebraic D-Modules." If this is ...
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1answer
239 views

Perverse sheaves (or D-modules) on vector spaces, constructible with respect to a hyperplane arrangement

Let $V$ be a finite dimensional complex vector space and let $\mathcal{A}$ be a finite collection of hyperplanes in $V$. Stratify $V$ by the intersections of elements of $\mathcal{A}$, and consider ...
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505 views

Recommendation textbooks on D-module

I am going to take part in a seminar on D-module and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, and A Primer of Algebraic D-Modules ...
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229 views

Holonomic ideals and D-finite power series

I would like to understand the connection between the term D-finite power series (in n variables) and the term of a holonomic module over the Weyl algebra A_n. A power series $f \in K[[x_1,...,x_n]]$ ...
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1answer
464 views

questions about coroot

I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the ...
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1answer
218 views

What theorem of Serre's is being referenced?

I am reading Joseph Bernstein's notes on D-modules which are available online here In section 1 (page 1 of the pdf) Bernstein writes "By Serre's theorem this condition is local." I was wondering to ...