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1answer
137 views

Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X \cong \oplus_a ...
5
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1answer
254 views

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 ...
3
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1answer
161 views

flat descent for perverse sheaves

Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$. Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
4
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1answer
179 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 ...
6
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1answer
213 views

Are there perverse sheaves on abelian varieties with small Euler characteristic?

Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can ...
3
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1answer
262 views

A question on algebraic loop groops

Setup: Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in ...
4
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1answer
127 views

Stalks of intersection cohomology complexes of Schubert varieties and Bruhat order

All varieties are over $\mathbb{C}$. Let $G$ be a connected reductive group, $B\subseteq G$ a Borel subgroup. Let $O_w$ be a $B$-orbit in $G/B$. I.e., $O_w$ is a Bruhat cell. In particular, it is ...
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0answers
104 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 $MHM(X)$ ...
7
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1answer
228 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 ...
6
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1answer
249 views

Iterated Milnor fibrations and Thom's a_f condition

Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question: Problem: Let ...
1
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2answers
168 views

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 ...
2
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1answer
264 views

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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0answers
102 views

Perverse sheaves for easy stratifications

Let $X$ be a complex variety equipped with a stratification. Let us assume, that all strata are contractible and in addition, that all strata closures are smooth. Is there an "easy" quiver ...
2
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0answers
122 views

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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0answers
163 views

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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0answers
129 views

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 ...
13
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1answer
594 views

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 ...
3
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1answer
220 views

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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0answers
101 views

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 ...
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1answer
214 views

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 ...
2
votes
1answer
270 views

Pullbacks of intermediate/middle extensions and Gabber's purity theorem

I am currently trying to understand intermediate extensions of perverse sheaves, specifically the proof of Gabber's purity theorem, which states that the intermediate extension of a pure perverse ...
2
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2answers
188 views

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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3answers
665 views

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?
4
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2answers
191 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 ...
5
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1answer
251 views

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 ...
14
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1answer
503 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 $\Delta_w$ ...
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0answers
97 views

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 ...
5
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0answers
365 views

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, ...
4
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1answer
298 views

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
votes
1answer
347 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 ...
4
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0answers
196 views

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' ...
0
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1answer
221 views

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 ...
4
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0answers
291 views

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 ...
2
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2answers
394 views

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/). ...
3
votes
1answer
226 views

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 $U$ ...
1
vote
1answer
165 views

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 ...
6
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0answers
222 views

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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0answers
192 views

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 ...
3
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0answers
331 views

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 ...
3
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0answers
270 views

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 ...
3
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0answers
367 views

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 ...
7
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1answer
402 views

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 ...
4
votes
1answer
300 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, ...
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1answer
516 views

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 ...
5
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1answer
306 views

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 ...
5
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1answer
454 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 ...
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2answers
1k views

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 ...
5
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0answers
493 views

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 ...
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0answers
2k views

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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1answer
592 views

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 ...