All Questions
Tagged with differential-operators d-modules
19 questions
28
votes
2
answers
2k
views
Codimension of the range of certain linear operators
Added:8/15/2024 What about holomorphic or real analytic version? Please see the comment discussions on this post.
Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We ...
- 356
17
votes
2
answers
1k
views
What is the motivation behind the characteristic variety of a D-module and what does it's geometry tell me about the D-module?
Given a smooth algebraic variety $X$, and an $\mathcal{M}\in \text{Mod}(D_X)$, there is the characteristic variety of $\mathcal{M}$ defined as
$$
\text{Char}(\mathcal{M}):= V\left(\sqrt{Ann(\mathcal{M}...
- 1,716
16
votes
2
answers
3k
views
The algebraic version of Riemann-Hilbert correspondence
It is well known that if I have a differentiable manifold (holomorphic manifold) $M$, then I have a functor from the category of vector bundles on $M$ with flat connections to the category of local ...
- 163
13
votes
1
answer
858
views
On the definition of regularity
In the literature on D-modules, there are many definitions of regularity of holonomic D-modules.
(1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its ...
- 7,527
11
votes
0
answers
614
views
Is the formal neighborhood of the diagonal a generalization of the Jet bundle?
Let $f: X \to S$ be a morphism of locally ringed spaces and $\triangle: X \to X \times_S X$ the corresponding diagonal morphism with kernel sheaf $\mathcal{I} = \ker \triangle^{\flat}$.
Definition: ...
- 111
7
votes
0
answers
250
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}_{\...
- 143
7
votes
0
answers
823
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 ...
- 21.8k
6
votes
1
answer
176
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[...
- 6,622
5
votes
1
answer
480
views
Regular holonomic D-modules as generalisation of regular singular points
I'm trying to understand why the definition of a regular holonomic D-module is a good generalisation of the usual definition of a regular singular point for a differential equation. More precisely, ...
- 1,017
5
votes
1
answer
541
views
Localizability of differential operators a la Grothendieck
Hello,
Maybe this question is trivial, so sorry
Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).
Then we can define the module of differential operators $D^{\leq n} (A)$,...
- 5,562
5
votes
0
answers
321
views
Weyl algebra acting on a polynomial ring
Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots,
x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl
algebra. As usual $W$ ...
- 51
3
votes
1
answer
198
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,$$
where $P$ is a linear ...
- 2,788
3
votes
0
answers
122
views
Analytic analogue of implicit functions for differential operators
Let $p\colon \mathbb{R}^2 \to \mathbb{R}$ be a polynomial with a non-vanishing gradient at $p^{-1}(0)$. Then, the implicit function theorem says that $S = \{(x,y) \in \mathbb{R}^2 \mid p(x,y) = 0\}$ ...
- 423
3
votes
0
answers
97
views
Transfer modules and Weyl algebra
Let $V$ be a $\mathbb{C}$-vectorial space of dimension $n$ and $V^*$ the complex dual space.
I would like to understand the following isomorphism $$D_{V^* \leftarrow V \times V^*} \overset{L}\otimes_{...
- 1,017
2
votes
1
answer
186
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 ...
- 5,701
2
votes
0
answers
123
views
How can I prove that this D-module is free?
I have the following setup, I expect that it is studied in the theory of $D$-modules, and I apologize in advance if I am wrong.
First, I have an algebra $A$ of differential operators on $n$ ...
- 1,488
1
vote
0
answers
61
views
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 ...
- 5,701
1
vote
0
answers
111
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 ...
- 2,788
0
votes
0
answers
241
views
Does regularity of a D-module for an unusual filtration imply regularity for the usual one?
One definition of regular D-modules on affine space is that a D-module is regular if it has a filtration compatible with the order filtration on differential operators whose associated graded is ...
- 44.7k