The perverse-sheaves tag has no usage guidance.

**2**

votes

**0**answers

97 views

### What sorts of weights for perverse sheaves were or can be computed?

I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would ...

**3**

votes

**0**answers

119 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 ...

**5**

votes

**0**answers

86 views

### Characteristic Cycles and Nearby Cycles

Let $Y$ be a smooth algebraic variety over $\mathbb{C}$, let $X = Y \times \mathbb{C}$ and let $f: X \to \mathbb{C}$ be the projection. Let $M$ be a (not necessarily regular) holonomic $D_X$-module ...

**3**

votes

**0**answers

161 views

### Deligne-Lusztig and Character sheaves

Consider: $G$ - a nice group ($GL_n$) over a finite field $F$. $X$ - the flag variety. Consider a nice $G$-equivariant $l$-adic sheaf $\mathcal{M}$ on $X \times X$, equipped with Weil structure.
Fix ...

**6**

votes

**1**answer

1k 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 ...

**0**

votes

**1**answer

66 views

### Decomposition of semi simple local systems

I found A question similar to this, but the answer wasn't clear to me and I'm not supposed to ask for further clarification in the answer section.
Let $L$ a semi simple local system defined over an ...

**1**

vote

**1**answer

142 views

### Relation between Milnor fiber and its restriction via vanishing cycles

I am reading these notes on nearby and vanishing cycles, where an initial assumption is made: the author talks about a complex analytic function $f:X\to \mathbb C$, assuming $X$ is closed in an open ...

**0**

votes

**1**answer

128 views

### cohomology of an intermediate extension of a local system

Let $V$ be affine $n$-space over a field $k$; and $j \colon U \to V$ an open subscheme of $V$. Let $L$ be an $\ell$-adic local system on $U$.
Suppose the cohomology of $H^{\bullet}(U,L)$ does not ...

**1**

vote

**2**answers

143 views

### Example to show that the inverse image under a finite morphism is not t-exact with respect to the perverse t-structure

According to Chapter 4 of Beilinson, Bernstein, and Deligne's "Faisceaux Pervers" (Asterisque 100, 1980) the inverse image $Rf^*$ with respect to a finite morphism $f$ is right t-exact with respect to ...

**2**

votes

**0**answers

194 views

### l-adic cohomology and perverse sheaves

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...

**6**

votes

**1**answer

338 views

### $\ell$-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 ...

**1**

vote

**1**answer

176 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 ...

**7**

votes

**1**answer

405 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**

votes

**1**answer

204 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**

votes

**1**answer

219 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**

votes

**1**answer

260 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**

votes

**1**answer

275 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**

votes

**1**answer

182 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 ...

**1**

vote

**0**answers

152 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**

votes

**1**answer

244 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**

votes

**1**answer

296 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**

vote

**2**answers

191 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**

votes

**1**answer

305 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 ...

**4**

votes

**0**answers

123 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**

votes

**0**answers

144 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 $X$ ...

**12**

votes

**0**answers

211 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 ...

**0**

votes

**0**answers

135 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**

votes

**1**answer

800 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**

votes

**1**answer

238 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 ...

**1**

vote

**0**answers

121 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 ...

**1**

vote

**1**answer

265 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

**1**answer

424 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**

votes

**2**answers

195 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 ...

**7**

votes

**3**answers

849 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?

**6**

votes

**3**answers

323 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**

votes

**1**answer

273 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**

votes

**1**answer

603 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 $\Delta_w$ ...

**0**

votes

**0**answers

107 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 ...

**6**

votes

**0**answers

494 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, ...

**5**

votes

**1**answer

387 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

**1**answer

383 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**

votes

**0**answers

202 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' ...

**1**

vote

**1**answer

312 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**

votes

**0**answers

362 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**

votes

**2**answers

422 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

**1**answer

254 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

**1**answer

179 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 ...

**7**

votes

**0**answers

238 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 ...

**0**

votes

**0**answers

219 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**

votes

**0**answers

394 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 ...