# Questions tagged [d-modules]

Modules over rings of differential operators.

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $S = \{(x,y) \in \mathbb{R}^2 \mid p(x,y) = 0\}$ ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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" (...
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### 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 $D$...
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### 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 ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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