All Questions
Tagged with perverse-sheaves d-modules
28 questions
30
votes
8
answers
4k
views
Applications of microlocal analysis?
What examples are there of striking applications of the ideas of microlocal analysis?
Ideally I'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...
- 7,789
14
votes
2
answers
1k
views
Relation between holonomic D-modules and perverse sheaves
Given a smooth complex algebraic variety, the Riemann-Hilbert-correspondence tells us, that the category of perverse sheaves is equivalent to the category of regular, holonomic D-modules.
However not ...
- 13.2k
11
votes
1
answer
837
views
How does one interpret the naive t-structure on constructible sheaves as a t-structure on D-modules?
By the Riemann-Hilbert correspondence, there is an equivalence between
(1)
$\mathcal{D}\operatorname{-mod}(X)$
, the (derived) category of holonomic D-modules on a complex variety X, and
(2)
...
- 3,351
10
votes
1
answer
1k
views
Computation of vanishing cycles
Here's the problem I'm looking at:
$F$ is a perverse sheaf (or a regular holonomic D-module, or even a mixed hodge module) on $\mathbb{C}^2$ stratified by $z_1 = 0$, $z_2=0$. It can be caracterized ...
- 7,527
9
votes
4
answers
3k
views
Gluing perverse sheaves?
It might be a stupid question.
How to glue perverse sheaves? I am considering the following example. Flag variety of $sl_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted by $...
- 5,515
9
votes
0
answers
446
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{...
- 5,901
7
votes
1
answer
884
views
Localization of vanishing cycles
Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function.
Question: Is it true that the $\lambda \in A^1$ ...
- 7,527
7
votes
0
answers
207
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-...
- 81
6
votes
1
answer
237
views
Additivity of characteristic cycle of holonomic D-module
Let $\mathcal{M}$ be a holonomic D-module on a complex analytic (or alternatively, algebraic) manifold $X$. One can attach to it (using a good filtration) a characteristic cycle $Ch(\mathcal{M})$ ...
- 21.8k
6
votes
0
answers
230
views
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 ...
- 71
6
votes
0
answers
225
views
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 ...
- 495
6
votes
0
answers
275
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 (...
- 272
6
votes
0
answers
233
views
Toric Degenerations and Nearby Cycles
Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...
- 1,543
6
votes
0
answers
714
views
Characteristic Cycles and Nearby Cycles
Let $Y$ be a smooth algebraic variety over $\mathbb{C}$, let $X = Y \times \mathbb{C}$ and let $f: X \to \mathbb{C}$ be the projection. Let $M$ be a (not necessarily regular) holonomic $D_X$-module ...
- 1,543
6
votes
0
answers
391
views
Blow ups and Characteristic varieties
Let $f:Y\to X$ be a morphism of smooth varieties (complex analytic or algebraic as you wish). We have
$$
T^* Y \overset{f_d}{\longleftarrow} Y\times_X T^*X \overset{f_\pi}{\longrightarrow} T^* X
$$...
- 7,527
5
votes
1
answer
317
views
Operations on perverse sheaves on disk
The category of perverse sheaves on the disk is isomorphic to the category of diagrams
$$E\substack{\substack{c\\\to}\\\substack{v\\\leftarrow}}F$$
With $E,V$ finite dimensional vector spaces, and ...
- 2,788
5
votes
1
answer
212
views
Riemann Hilbert Correspondence with fixed stractification
Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category ...
- 677
4
votes
1
answer
241
views
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 ...
- 371
4
votes
1
answer
437
views
Smooth mixed hodge modules - representations of fundamental group?
I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...
- 5,562
4
votes
0
answers
117
views
Is there an "$\ell$-adic Riemann Hilbert correspondence"?
The Riemann-Hilbert correspondence (see, e.g., Thm. 7.2.2 of D-modules, perverse sheaves, and representation theory) shows that analytic perverse sheaves are equivalent to regular holonomic $D$-...
- 615
4
votes
0
answers
200
views
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 ...
- 121
4
votes
0
answers
344
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 ...
- 5,701
4
votes
0
answers
290
views
The associated graded of a mixed Hodge module
Unfortunately this question will be a bit vague, since the question revolves around a memory of something I may have heard in a talk (long time ago).
Let $X$ be a smooth complex variety. Let $MHM(X)$ ...
- 1,595
3
votes
1
answer
667
views
Polarizable variations of (mixed) Hodge structures
I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be ...
- 1,595
3
votes
1
answer
282
views
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{...
- 711
3
votes
0
answers
76
views
Is the characteristic cycle map for perverse sheaves injective?
Let $X$ be a smooth irreducible complex variety. Is the characteristic cycle map from the Grothendieck group of perverse sheaves (with complex coefficients) on $X$ to the free abelian group generated ...
- 333
2
votes
0
answers
112
views
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?
- 507
1
vote
0
answers
172
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 $*$ ...
- 3,019