Questions tagged [d-modules]

Modules over rings of differential operators.

Filter by
Sorted by
Tagged with
1 vote
1 answer
99 views

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^\...
user avatar
  • 1,317
4 votes
0 answers
110 views

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,...
user avatar
  • 794
3 votes
1 answer
113 views

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 ...
user avatar
  • 2,655
2 votes
1 answer
129 views

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 ...
user avatar
  • 4,004
5 votes
1 answer
207 views

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 ...
user avatar
5 votes
1 answer
237 views

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 ...
user avatar
  • 1,317
4 votes
0 answers
93 views

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 ...
user avatar
  • 1,675
4 votes
1 answer
252 views

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_*:\...
user avatar
  • 1,317
3 votes
1 answer
242 views

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 ...
user avatar
  • 19k
7 votes
0 answers
172 views

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-...
user avatar
2 votes
1 answer
162 views

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 ...
user avatar
3 votes
0 answers
188 views

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 ...
user avatar
  • 794
3 votes
1 answer
178 views

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 ...
user avatar
  • 2,921
4 votes
1 answer
293 views

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}...
user avatar
  • 1,317
5 votes
1 answer
186 views

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....
user avatar
  • 1,121
6 votes
0 answers
248 views

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 ...
user avatar
  • 1,317
6 votes
0 answers
228 views

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 (...
user avatar
  • 273
3 votes
0 answers
102 views

"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)...
user avatar
  • 2,655
7 votes
2 answers
595 views

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 ...
user avatar
1 vote
0 answers
122 views

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 $*$ ...
user avatar
  • 2,640
7 votes
0 answers
272 views

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) $M\otimes_{\mathcal{O}_X} N$...
user avatar
  • 1,317
9 votes
0 answers
251 views

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 ...
user avatar
  • 41.2k
3 votes
1 answer
230 views

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, ...
user avatar
  • 273
2 votes
0 answers
84 views

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,...
user avatar
  • 81
2 votes
0 answers
77 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 ...
user avatar
  • 81
3 votes
0 answers
223 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 ...
user avatar
  • 4,004
5 votes
0 answers
156 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, \...
user avatar
  • 2,640
4 votes
1 answer
267 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 ...
user avatar
  • 4,926
8 votes
0 answers
312 views

$K$-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{...
user avatar
  • 4,926
5 votes
1 answer
148 views

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) ...
user avatar
  • 1,703
3 votes
1 answer
199 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 ...
user avatar
  • 19k
3 votes
1 answer
169 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 ...
user avatar
  • 2,655
3 votes
0 answers
126 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}\...
user avatar
  • 4,004
5 votes
0 answers
164 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 ...
user avatar
  • 1,675
2 votes
0 answers
59 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 ...
user avatar
4 votes
0 answers
82 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 $\...
user avatar
3 votes
0 answers
68 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 ...
user avatar
  • 123
3 votes
0 answers
279 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 ...
user avatar
  • 4,004
0 votes
1 answer
186 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, ...
user avatar
7 votes
0 answers
244 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}_{\...
user avatar
  • 143
4 votes
0 answers
183 views

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,...
user avatar
  • 1,675
4 votes
0 answers
96 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}\...
user avatar
4 votes
0 answers
89 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 ...
user avatar
11 votes
0 answers
220 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\...
user avatar
  • 3,472
1 vote
0 answers
97 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 ...
user avatar
  • 2,655
5 votes
1 answer
515 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}...
user avatar
  • 1,703
2 votes
0 answers
131 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 ...
user avatar
  • 2,655
6 votes
0 answers
134 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, $\...
user avatar
13 votes
2 answers
619 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 ...
user avatar
  • 4,004
3 votes
0 answers
98 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 ...
user avatar
  • 183

1
2 3 4 5