Questions tagged [perverse-sheaves]
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206 questions
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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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2
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270
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on a characterisation of the intersection complex
Let $X$ be an integral scheme of finite type over a field $k$ of dimension $d$ and $U$ an open dense smooth subscheme.
Let $K\in D_{c}^{b}(X,\bar{\mathbb{Q}}_{l})$ be such that $K_{U}=\bar{\mathbb{Q}}...
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1
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Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors
I hope this question is not too vague.
Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$.
Denote by $\pi:G/B\to G/P$ the canonical map. Consider ...
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what does the decomposition theorem say for a Lefschetz pencil?
The setting is the following: let $X$ be a smooth projective variety (say over $\mathbb{C}$), $D$ a simple normal crossings divisor on $X$ and $(H_t)_{t \in \mathbb{P}^1}$ a Lefschetz pencil on $X$ ...
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Constructible derived category and fundamental category
Introduction (may be skipped)
Given a nice topological space $X$, the category of local systems (say over a field $k$) on it is equivalent to the category of representations of its fundamental ...
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0
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0
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166
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The intersection complex and the Cohen-Macaulay property
Let $\Delta:Y\rightarrow X$ a closed immersion of $k$-schemes of finite type and equidimensionnal.
We assume that $\Delta^{*}[-d]IC_{X}=IC_{Y}$, if $X$ is Cohen-Macaulay, does it imply that $Y$ is ...
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1
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What is the purpose of section 3 of BBD?
I am not quite sure that this question is appropriate for Mathoverflow, yet I would be deeply grateful for any hint: what happens in section 3 of Beilinson A., Bernstein J., Deligne P., Faisceaux ...
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3
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1
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is this intersection complex a sheaf?
Let $X$ be a smooth complex projective variety and $D$ a normal crossing divisor. Assume that you are given a local system $V$ of complex vector spaces on $X-D$ having finite monodromy. Consider the ...
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on geometric Satake and functions
Let $G(F)/G(O)$ the affine grassmanian with $F=k((t))$ where $k$ is a finite field.
For $\lambda$ a dominant cocharacter, we have by Cartan decomposition the schubert strata $\overline{Gr^{\lambda}}$....
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1
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Decomposing Semisimple Perverse Sheaves
So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do ...
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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 ...
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3
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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?
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1
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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 ...
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10
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1
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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 ...
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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
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1
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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 ...
- 3,806
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1
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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 ...
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1
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1
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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 ...
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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
3
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1
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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 $U$ ...
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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-...
CommunityBot
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2
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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/).
...
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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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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 ...
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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 (...
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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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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 ...
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5
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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 ...
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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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10
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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 ...
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3
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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 ...
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8
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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....
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8
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1
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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 ...
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1
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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, ...
- 1,576
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1
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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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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 ...
CommunityBot
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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 ...
CommunityBot
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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
$$
T^* Y \overset{f_d}{\longleftarrow} Y\times_X T^*X \overset{f_\pi}{\longrightarrow} T^* X
$$...
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2
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1
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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 ...
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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 ...
CommunityBot
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1
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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 ...
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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)...
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2
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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 = ...
- 1,576
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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 $S$. ...
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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 =...
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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, ...
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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)
...
- 2,083
5
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1
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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 ...
CommunityBot
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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.
- 35.2k
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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 ...
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