All Questions
10 questions
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
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
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
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