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Questions tagged [perverse-sheaves]

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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/...
Saal Hardali's user avatar
  • 7,789
7 votes
0 answers
602 views

What's the relationship between the different versions of the BBD decomposition theorem?

I have a few questions relating to the BBD decomposition theorem. I have come across the following two versions of the decomposition theorem. Version 1. Let $f : X \to Y$ be a proper map of ...
Balerion_the_black's user avatar
22 votes
4 answers
5k views

Examples for Decomposition Theorem

There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne. Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...
Ilya Nikokoshev's user avatar
17 votes
1 answer
7k views

A nice explanation of what is a smooth (l-adic) sheaf?

I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.
Mikhail Bondarko's user avatar
14 votes
1 answer
3k views

When does a perverse sheaf occur in the decomposition theorem?

Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...
Tian An's user avatar
  • 3,799
14 votes
1 answer
1k views

Morphisms between Verma modules

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ ...
Reladenine Vakalwe's user avatar
8 votes
1 answer
530 views

Restriction to Levi Subgroups and the Affine Grassmannian

Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$. My Question: What is the geometric analogue of the restriction ...
Oliver Straser's user avatar
7 votes
1 answer
2k views

References on semismall maps

Where can I find references on semismall maps, in the sense of Goresky and MacPherson? I don't want to restrict to the case where the base is $\mathbb C$ (an arbitrary alg. closed field would be fine),...
shenghao's user avatar
  • 4,265
5 votes
1 answer
535 views

Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves

I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, ...
Mikhail Bondarko's user avatar
5 votes
1 answer
508 views

Are equivariant perverse sheaves constructible with respect to the orbit stratification?

[Moved here from MSE] Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$. Question. Is it true ...
W.Rether's user avatar
  • 455
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 ...
user2520938's user avatar
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4 votes
1 answer
291 views

Convolution of $\ell$-adic sheaves is commutative if the group is commutative

[This is a duplicate of this question on Stackexchange] I am trying to figure out how to prove a very basic statement about convolution of $\ell$-adic/perverse sheaves in Katz's "Rigid local systems" ...
Yoël's user avatar
  • 329
3 votes
1 answer
251 views

About decomposition theorem BBD with respect to some stratification

I want to follow up a question from here (how to deduce version 1.a. from version 1). I know a version of decomposition theorem BBD: Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex ...
Toan's user avatar
  • 133
2 votes
1 answer
179 views

Intermediate extension and perverse cohomologies

Let a set X be the union of two locally closed subsets U and V such that U does not lie in the closure of V. Let the restriction of a complex R of constructible sheaves on X to a smooth open subset A ...
Yellow Pig's user avatar
  • 2,964
2 votes
1 answer
218 views

Intermediate extension and irreducible subquotients of perverse cohomologies

Let a set X be the union of two locally closed subsets U and V such that U does not lie in the closure of V. Let the restriction of a complex R of constructible sheaves on X to a smooth open subset A ...
Yellow Pig's user avatar
  • 2,964
1 vote
2 answers
279 views

Correspondences acting on cohomology groups $H^*(X)$ & splittings

Let $X$ be a smooth connected proper scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $...
JackYo's user avatar
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