Questions tagged [perverse-sheaves]
The perverse-sheaves tag has no usage guidance.
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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8
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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/...
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27
votes
0
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959
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Nearby cycles without a function
Suppose that:
$X$ is a smooth complex algebraic variety,
$f : X \to D$ is a proper map to a small disc, smooth away from 0,
$Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$.
Then there is a procedure (...
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24
votes
2
answers
4k
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Why is the decomposition theorem awesome?
I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...
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2
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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 ...
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22
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4
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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 ...
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20
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3
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So what exactly are perverse sheaves anyway?
Is there a way to define perverse sheaves categorically/geometrically? Definitions like the following from lectures by Sophie Morel:
The category of perverse sheaves on $X$ is $\mathrm{Perv}(X,F):=...
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0
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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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17
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1
answer
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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.
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Gabber's original proof of his purity theorem
Gabber's purity theorem is the statement that if $\mathscr{F}$ is a pure perverse sheaf on an open subvariety $j : U \hookrightarrow X$ then so is $j_{!*} \mathscr{F}$.
It is remarkable because it ...
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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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15
votes
1
answer
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IC sheaf of certain explicit variety
Let $n,m$ be two positive integers. Let $Z$ denote the closed subvariety
in $\mathbb A^n \times \mathbb A^m$
given by the equation $x_1...x_n=y_1...y_m$.
QUESTION: What is the stalk (with the action ...
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0
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Applications of character sheaves
There are many important recent works (for example, by Lusztig, Bezrukavnikov-Finkelberg-Ostrik, Ben-Zvi-Nadler, Boyarchenko-Drinfeld, Lusztig-Yun, Vilonen-Xue) on character sheaves (which are certain ...
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3
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Counterexamples to gluing complexes of sheaves
Note: I asked the question below last week on MathSE but received no answer.
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This ...
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1
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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 ...
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2
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"Correct" definition of stratified spaces and reference for constructible sheaves?
It seems that the theory of constructible sheaves (in particular anything that goes into proving that they form an abelian category) requires some technical statements about existence of certain ...
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2
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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 ...
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14
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2
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Non semi-simple monodromy in an algebraic family
I am looking for an example of a (edit: projective) family
$f : X \to Y$
of complex algebraic varieties which is a topologically locally trivial fibration in (singular) varieties and such that there ...
- 6,162
14
votes
1
answer
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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 $\Delta_w$ ...
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4
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How to do Computations Using the Decomposition Theorem for Perverse Sheaves
This is a follow-up to this post on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.
My question is how does one use the ...
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0
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Kakuro puzzles and sheaf cohomology
This is a recreational, summer question and could be more well-suited for mathstackexchange. However, some of you on holiday could appreciate the topic. I recently came across Kakuro Puzzles, similar ...
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1
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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, ...
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12
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0
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Perverse sheaves and representation theory
At my university we are having a working group on perverse sheaves, with the aim of applying them to representation theory (Lusztig canonical bases for quivers/quantum groups etc). We are still ...
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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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11
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3
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Examples of calculating perverse sheaves on algebraic varieties with easy stratification
I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
http://www.amazon.com/Introduction-...
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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)
...
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11
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2
answers
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What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?
In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...
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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 ...
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10
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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 ...
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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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10
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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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intuition about perverse sheaves
firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...
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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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A recommendation for a book on perverse sheaves
I would like to learn about perverse sheaves.
I will be grateful if someone could recommend me a book with the following structure.
Introduction to basic homotopy theory (derived category and t-...
9
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4
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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 $...
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$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{...
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0
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On the definition of triangulated categories
Triangulated categories were introduced in the 1960s by Grothendieck and Verdier in order to develop homological algebra in the framework of derived categories. An example of a triangulated category ...
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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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8
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1
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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$6....
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$\ell$-adic monodromy theorems (over $\mathbb{C}$)
This question is about $\ell$-adic monodromy theorems for families over a number field. ($\ell$-adic analogues of Corollaries 6.2.8 and 6.2.9 in [BBD].)
Notation
$H$ denotes étale cohomology.
Let $...
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1
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530
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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 ...
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8
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1
answer
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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
answer
714
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DG enhancements of $\ell$-adic derived categories
This question is similar in flavor to Existence of dg realization for 6 functors
Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...
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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 $S$. ...
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What is an example of Beilinson's theorem on $D^b\mathrm{Perv}$ failing for non-field coefficients?
In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $...
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Beilinson's theorem for fixed stratifications
Beilinson's theorem states that for a variety $X$ and a field $k$ the realization functor
$$\text{real}: D^b\text{Perv}(X,k)\to D_c^b(X,k)$$ is an equivalence of categories.
If we only consider ...
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What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
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473
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Lefschetz pencils and perverse sheaves
I have been reading BBD and Geordie Williamson's “An illustrated guide to perverse sheaves”. The latter has been tremendously helpful to get some intuition for the former.
Let $K$ be some field, and ...
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980
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What is an example of a non-mixed $\ell$-adic sheaf?
$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...
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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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