Questions tagged [perverse-sheaves]
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206 questions
2
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2
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A submodule of a constant D-module is constant
Hello,
Let $X$ be a smooth variety in char. 0. Let us call a $D$-module on $X$ constant, if it is isomorphic to a finite direct sum of the $D$-modules $O$ (the sheaf of regular functions with the ...
- 5,562
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votes
3
answers
2k
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Applications for intersection (co)homology and for the Decomposition Theorem for students?
Which applications of intersection (co)homology and of the (Topological) Decomposition Theorem have most chances to be understood by students?
- 16.9k
7
votes
3
answers
716
views
Nice algebraic approximations of classifying spaces
Let $G=GL_k(\mathbb C)$ be the complex linear group. Then the infinite Grassmannian is a model for the classifying space $BG$.
We can write the infinte Grassmannian as a colimit of the finite ...
- 13.2k
6
votes
1
answer
504
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Geometric interpretation of translation through the wall
What does translation through the wall correspond to under Beilinson Bernstein localization?
More precisely I am interested in the following:
There is a well known equivalence between the principal ...
- 13.2k
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 $\Delta_w$ ...
- 1,595
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0
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150
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descent of a complex of sheaves
Let X a projective variety over an algebraically closed field on which an abelian variety $A$ acts freely.
Then $X/A$ is a projective variety. Let $f:X\rightarrow X/A$
Let $K\in D_{c}^{\leq 0}(X,\bar{...
- 3,472
12
votes
1
answer
1k
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On the derived category of constructible étale sheaves
The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible.
Clearly, ...
- 15.4k
7
votes
1
answer
966
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Geometric intuition behind perverse coherent sheaves?
I would like to know an intuition behind perverse coherent sheaves. I am aware that it is induced by a heart of another t-structure on the derived category. Are there any better, probably more ...
- 71
3
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1
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667
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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
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0
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281
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Mixed structures on Hom spaces induced by mixed sheaves
Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let
$rat\colon D^b_m(X)\to D^b(X)$
be the `forgetful' ...
- 1,595
1
vote
1
answer
1k
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Intermediate extension functor exact?
It is well known, that the intermediate extension functor $j_{!*}$ preserves injections and surjections. However it seems that it is not exact in general!
1) What would be an example which shows that ...
- 13.2k
5
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0
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564
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About an argument in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel.
I am trying to understand Proposition 3.4.2 in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel [BGS]. A copy of the paper can be found at http://home.mathematik.uni-...
- 1,595
2
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2
answers
655
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Question regarding a statement in `A proof of Jantzen conjectures'
So I am trying to understand a statement in the proof of Corollary 5.2.3 in `A proof of Jantzen conjectures' (a copy of the paper can be found at http://www.math.harvard.edu/~gaitsgde/grad_2009/).
...
- 1,595
3
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1
answer
418
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How to glue perverse sheaves of abelian groups?
Let $X$ be a complex algebraic variety and consider the category $P(X)$ of perverse sheaves of complex vector spaces.
Let $f:X\rightarrow \mathbb C$ be a regular function, $Z$ its zero set and $U$ ...
- 13.2k
1
vote
1
answer
243
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How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?
Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...
- 16.9k
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Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?
The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it (...
- 16.9k
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0
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253
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Online reference for bridge between $\mathbb C$ and $\mathbb F$
I am looking for a text which
1) Explains how to deduce statements about perverse sheaves on complex geometry from analogous statements in positive characteristic. For example the last chapter "De F ...
- 13.2k
4
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0
answers
540
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Geometric picture behind tilting sheaves
I am trying to read "Tilting exercises" and have trouble to see any geometric pictures behind the formulas.
So my questions are, how to think about tilting perverse sheaves?
Are they just formal ...
- 13.2k
5
votes
0
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491
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Are Lusztig's perverse sheaves the only equivariant ones with nilpotent characteristic cycle?
In his '91 paper, Lusztig defines a collection of simple perverse sheaves that correspond to the canonical basis; these are defined using a pushforward construction, and from the definition, it's easy ...
- 44.7k
3
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0
answers
516
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Schubert varieties of flag variety , perverse sheaves
The set of Schubert varieties in a flag variety is in one-to-one correspondence with elements of the Weyl group via left cells. There is also some relation between products of Schubert varieties and ...
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1
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983
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l-adic vs complex Perverse Sheaves
Let $X$ be a scheme of finite type over $Spec(\mathbb{C})$. Let $X_{an}$ denote the associated complex analytic space. After fixing an isomorphism $\overline{\mathbb{Q}}_l\cong \mathbb{C}$, by $\S$6....
- 491
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535
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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, ...
- 16.9k
10
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1
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843
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Crystalline analogue of perverse sheaves
Consider a variety $X$ over a field $k$ and let $\ell$ be a prime different from the characteristic of $k$. One has the derived category $D(X, Q_{\ell})$ of $\ell$-adic sheaves. There are very ...
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826
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Parabolic convolution of perverse sheaves in terms of the Hecke algebra
It is "well-known" that the Hecke algebra $\mathcal{H}$ can be thought
of as the Grothendieck group for the category of perverse sheaves on
$G/B$, where the product in $\mathcal{H}$ corresponds to ...
- 3,093
7
votes
1
answer
2k
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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),...
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votes
2
answers
5k
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What exactly does the weight filtration in Hodge theory have to do with the Weil conjectures?
Let $X$ be a variety over $\mathbb{C}$, say separated. According to Deligne's results, there is a "mixed Hodge structure" on the total cohomology $H^\bullet(X(\mathbb{C}), \mathbb{Z})$. One component ...
- 25.6k
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735
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Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?
This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
- 16.9k
32
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0
answers
3k
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Microlocal geometry - A theorem of Verdier
(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...
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1
answer
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Bad behaviour of perverse sheaves over 'general' bases?
Could one define $\mathbb{Q}_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties ...
- 16.9k
1
vote
1
answer
431
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Is it easy to define weights for $Q_l$-sheaves over finite type $Z[1/l]$-schemes?
In her paper "Mixed perverse sheaves for schemes over number fields" A. Huber defines certain weights for certain categories of $\mathbb{Q}_l$-sheaves over a finite type $\mathbb{Q}$-scheme $...
- 16.9k
2
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1
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528
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Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 16.9k
6
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0
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391
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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
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1
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434
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How would you call the 'base' of a (intermediate extension of) perverse sheaf?
Let $j:U\to S$ be an (open) immesrion; let $P_U$ be a perverse (\'etale, though my question makes sense in the topological setting also) sheaf on $U$. Then I would like to say that
the intermediate ...
- 16.9k
3
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1
answer
767
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Is there a 'classical' definition for the support of a perverse sheaves.
I would like to define the support of a mixed motivic sheaf. This should be something similar to the support of a perverse sheaf.:) Is there any 'classical' definition for the latter?
I suspect that ...
- 16.9k
1
vote
0
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765
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Which statement do people usually call the Decomposition Theorem, and what is the precise reference for it?
Which statement is usually called the Decomposition Theorem (for perverse sheaves)? Is this (roughly): a proper pushforward of an intersection complex could be decomposed into a direct sum of (shifted)...
- 16.9k
8
votes
1
answer
729
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The conjectural relation between mixed motivic sheaves and the perverse t-structure.
As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. ...
- 16.9k
5
votes
3
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921
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Reference for two facts about perverse sheaves on G/B
I wonder whether there is a reference for the following two things:
The Grothendieck group of B-equivariant semisimple? perverse sheaves on $G/B$ is the Hecke-algebra.
The category of B-equivariant ...
- 13.2k
14
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2
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1k
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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
6
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1
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651
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Intersection Cohomology of Coordinate Hyperplanes
I'm trying to learn how to compute stalks of IC sheaves, and I was wondering about the following example:
Fix $n$. Let $X \subset \mathbb{C}^n$ be the variety cut out by the equation $x_1 \cdots x_n =...
- 5,548
1
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1
answer
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intermediate/middle extension of perverse sheaves
Does anybody know references for perverse sheaves, especially the intermediate/middle extension functor for $\mathbf{Q}_\ell$-sheaves for varieties over (the algebraic closure of) finite fields, ...
- 11
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1
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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
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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
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1
answer
765
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Easy special cases of the decomposition theorem?
The decomposition theorem states roughly, that the pushforward of an IC complex,
along a proper map decomposes into a direct sum of shifted IC complexes.
Are there special cases for the decomposition ...
- 13.2k
17
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1
answer
7k
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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.
- 16.9k
7
votes
2
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702
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Which statements in section 5 of BBD will fail if we consider $\mathbb{Q}_l$-adic sheaves there?
A stupid question: which statements in section 5 of BBD will fail if we replace $\overline{\mathbb{Q}_l}$-sheaves by just $\mathbb{Q}_l$-ones? I am especially interested in Proposition 5.1.15.
BBD = ...
- 16.9k
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2
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In what setting does one usually define mixed sheaves and weights for them?
In BBD mixed sheaves and weights for them were only defined for ($\overline{\mathbb{Q}_l}$-)sheaves over a variety $X_0$ defined over a finite field $F$. Weights start to behave better when one ...
- 16.9k
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3
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2k
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What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwise pure?
I'll freely admit that I have a rather hard time keeping straight different notions of purity of etale sheaves, and I think part of the problem is the lack of counterexamples.
For example, it's a ...
- 44.7k
9
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4
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3k
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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
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1
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792
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What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?
Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see his paper) corresponding to canonical basis vectors.
I'm interested in the stalks ...
- 44.7k
4
votes
1
answer
639
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Morphisms between pure complexes of sheaves
I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...
- 16.9k