Questions tagged [d-modules]

Modules over rings of differential operators.

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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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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_*(\...
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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 ...
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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?
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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 $[n]_q=(q^{n}-1)/(q-1)$ ...
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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. ...
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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. ...
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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{...
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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 ...
Martin Tang's user avatar
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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 ...
user577413's user avatar
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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 ...
Pulcinella's user avatar
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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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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, \...
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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, ...
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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 ...
Exit path's user avatar
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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....
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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\}...
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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^\...
Gabriel's user avatar
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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{...
Gabriel's user avatar
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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 $...
Exit path's user avatar
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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 ...
Rami's user avatar
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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^\...
Gabriel's user avatar
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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,...
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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 ...
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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 ...
Pulcinella's user avatar
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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 ...
Fernando Peña Vázquez's user avatar
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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 ...
Gabriel's user avatar
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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 ...
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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_*:\...
Gabriel's user avatar
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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 ...
asv's user avatar
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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-...
Rgdn Dznrbx's user avatar
2 votes
1 answer
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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 ...
curious math guy's user avatar
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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 ...
XT Chen's user avatar
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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 ...
Avi Steiner's user avatar
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4 votes
1 answer
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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}...
Gabriel's user avatar
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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....
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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 ...
Gabriel's user avatar
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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 (...
Mathmank's user avatar
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"higher" micro-support

Recall that for a sheaf $F$ on an analytic manifold $X$ the micro-support consists of those $\omega\in T^*X$ for which there exists a $C^1$ function $f$ defined around $\pi(\omega)$ with $f(\pi(\omega)...
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Why is the ring of Grothendieck differential operators bad when $X$ is singular?

$\DeclareMathOperator\Diff{Diff}$Suppose for simplicity that $X$ is affine, it is then possible to define $\Diff(X)$ — the ring of Grothendieck differential operators. When $X$ is smooth, then ...
FunctionOfX's user avatar
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When does a $D$-module think it’s a pullback along a smooth morphism?

Let $X$ and $Y$ be two algebraic varieties, and let $f: X \to Y$ be a morphism. Suppose $A$ is a holonomic $D$-module on $Y$. In this situation we can pull $A$ back to $X$ using either the $!$ or $*$ ...
Exit path's user avatar
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Tensor-hom adjunction for $\mathcal{D}$-modules

Let $X$ be a smooth equidimensional scheme over $\mathbb{C}$. Given two left $\mathcal{D}_X$-modules $M$ and $N$, we endow their tensor product (as $\mathcal{O}_X$-modules) $M\otimes_{\mathcal{O}_X} N$...
Gabriel's user avatar
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9 votes
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Understanding the mixed Hodge structure on D-modules on $\mathbb{C}$

Consider the category of regular D-modules on $\mathbb{C}$ (let's say as a complex manifold). It's a well-known fact that if you consider the subcategory of these where the singular support is the ...
Ben Webster's user avatar
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Iterating specialization of sheaves?

This is a question about the operation of taking the specialization of sheaves along a subspace. I'll recall the settings in which I've encountered a notion of specialization of sheaves: The real, ...
Mathmank's user avatar
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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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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 ...
AK12N1's user avatar
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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 ...
Pulcinella's user avatar
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5 votes
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198 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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