Questions tagged [perverse-sheaves]

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1 answer
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Isomorphic IC sheaves induced from different locally closed subvarieties

Let us work with varieties over $\mathbb{C}$ and $D^{b}_{c}(X)$ the bounded constructible derived category of sheaves of $\mathbb{Q}$ vector spaces. Say $X$ and $Y$ are smooth locally closed ...
1 vote
0 answers
153 views

Are there any relations between perverse t-structure (cohomologies) and standard t-structure (cohomologies)?

I'm reading the Corollary 3.2.3. in Exponential motives by J. Fresan and P. Jossen. The authors use the following statement in the proof of Corollary 3.2.3: let $C$ be any object in the derived ...
5 votes
1 answer
302 views

Operations on perverse sheaves on disk

The category of perverse sheaves on the disk is isomorphic to the category of diagrams $$E\substack{\substack{c\\\to}\\\substack{v\\\leftarrow}}F$$ With $E,V$ finite dimensional vector spaces, and ...
4 votes
0 answers
139 views

$\pm 1$-equivariant perverse sheaves on the affine line

Let $G=\mathbb{Z}/\mathbb{2Z}$ act by the map $z\mapsto -z$ on a complex line $\mathbb{C}$. The category $\mathcal{Perv}(\mathbb{C})$ of perverse sheaves smooth along the stratification by the origin ...
4 votes
1 answer
194 views

Are perverse sheaves representations of some topological invariant?

The well-known correspondence between vector bundles with flat connection on a smooth complex algebraic variety $X$ and complex representations of $\pi_1(X^{an})$, the fundamental group of the ...
2 votes
0 answers
142 views

Do the nearby cycle and Beilinson's vanishing cycle functors commute?

Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
5 votes
1 answer
553 views

Intersection cohomology and Poincaré duality

When trying to learn about perverse sheaves I hand-wavingly thought that intersection cohomology is the ‘minimal’ way of fixing the failure of Poincaré duality. But I am very aware that it is risky to ...
4 votes
0 answers
162 views

How to think about Beilinson's gluing data?

Let $X$ be a complex manifold, $D$ a divisor (that is globally the zero locus of a function) and $U$ its complement. Recall Beilinson's "how to glue perverse sheaves": Given a perverse ...
4 votes
0 answers
92 views

Simpson correspondence for perverse sheaves

Let $X$ be a projective complex manifold. Then Simpson's correspondence from nonabelian Hodge theory shows that the category of semisimple local systems on $X$ is equivalent to the category of ...
2 votes
0 answers
89 views

Applications of the Riemann-Hilbert Correspondence

I am aware of the (first) proof of the Kazhdan–Lusztig conjectures using the Riemann-Hilbert Correspondence. Are there any other interesting applications of the RH correspondence?
1 vote
0 answers
48 views

Cohomology of an intermediate extension (perverse) sheaf on the affine line

Let $\mathbb{A}^1$ be defined over a finite field or $\mathbb{C}$, $j: \mathbb{G}_m \rightarrow \mathbb{A}^1$ and $\mathcal{F}$ a local system on $\mathbb{G}_m$. I wonder what is known about the ...
1 vote
1 answer
324 views

Tensor product and semisimplicity of perverse sheaves

Let $X/\mathbb{C}$ be a smooth algebraic variety. Let $D_c^b(X,\bar{\mathbb{Q}}_{\ell})$ be the category defined in 2.2.18, p.74 of "Faisceaux pervers" (by Beilinson, Bernstein and Deligne). ...
2 votes
0 answers
124 views

Comparison of IC sheaves on Schubert varieties on two settings (l-adic vs. complex)

This question is basically about comparison of IC sheaves (or their sheaf cohomologies) for the settings: 1. variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear, 2. variety is over a finite ...
12 votes
1 answer
1k 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, ...
7 votes
1 answer
413 views

Equivariant perverse sheaves and orbit stratification

Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$. The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
2 votes
0 answers
114 views

Canonical basis and perverse coherent sheaves on the nilpotent cone

In the paper of Ostrik, he introduced a canonical basis of $K^{G\times {\mathbb C}^*}(\mathcal N)$, where $\mathcal N$ is the nilpotent cone for the group $G$. Question: does this canonical basis ...
8 votes
0 answers
317 views

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 ...
2 votes
1 answer
324 views

Extending IC sheaves across smooth divisors with normal crossings

I am trying to understand paragraph 1.6 of Lusztig's paper "Character Sheaves I". The basic setup is that $X$ is a smooth irreducible variety over a field $k=\overline{k}$, $D_i, i=1,...,r$ ...
4 votes
0 answers
180 views

D-modules generated by derivatives of Delta function

We consider D-modules over the affine line $\mathbb{A}_{\mathbb{C}}^{1}$, i.e. modules of Weyl algebra $A_1(\mathbb{C})=\mathbb{C}[x,\partial]$. There is a set of microfunctions (ref. A Primer of ...
1 vote
2 answers
238 views

Correspondences acting on cohomology groups $H^*(X)$ & splittings

Let $X$ be a smooth connected proper scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $...
27 votes
0 answers
921 views

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 (...
1 vote
1 answer
158 views

$\text{Ext}$-groups of perverse sheaves with a fixed stratification

Let $X$ be a complex variety with a good stratification $S$ and consider the category $Perv_S(X)$ of sheaves perverse with respect to the given stratification (with middle perversity) lying in $D^b_S(...
3 votes
0 answers
291 views

Understanding the proof of the Springer correspondence

Let $G$ be a connected reductive group over an algebraically closed field $k$ with Weyl group $W$. Let $$ \mathcal{S} = R\pi_*\mathbb{Q}_\ell[\dim \mathcal{N}] $$ be the Springer sheaf, where $\...
4 votes
1 answer
129 views

Explicit description of perverse sheaves on a disk

In How to glue perverse sheaves Beilinson claims that the category of perverse sheaves on the complex unit disk $D$ with the stratification with the closed strata $\{0\}\subset D$ is equivalent to the ...
10 votes
2 answers
1k views

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 ...
2 votes
0 answers
180 views

Springer sheaf and Deligne-Lusztig induction

Let $G=Gl_n$ be the general linear group over the algebraic closure of a finite field $\overline{\mathbb{F}}_q $ and let $F:G \to G$ be the standard Frobenius. On $G$ there is the Springer (perverse) ...
4 votes
0 answers
294 views

Perverse sheaves with stable infinity categories

I hope this question is not too naive. I have recently been trying to get familiar with the theory of stable $\infty$-categories. Lurie's Higher Algebra explains that they are a useful 'upgrade' of ...
3 votes
1 answer
292 views

Perverse tilting sheaves

In the article titled Tilting Exercises (See http://arXiv.org/abs/math/0301098v3) the authors define a notion of tilting perverse sheaves on an algebraic vareity $X$ with respect to stratification $\...
1 vote
1 answer
424 views

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 $...
2 votes
0 answers
80 views

Hard Lefschetz for perverse sheaves on Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold, $k\ge0$, $P\in Perv(X)$ be a semisimple object, then do we have the hard Lefschetz isomorphism between perverse cohomology sheaves $\omega^k:{}^p\mathcal{...
2 votes
0 answers
76 views

Restricting perverse intermediate extension to closed complement

Consider a scheme $X$ over athe complex numbers, $j:U\to X$ an open subscheme, $i:Z\to X$ its closed complement, and a perverse sheaf $F$ over $U$ with complex coefficients. The intermediate extension ...
5 votes
3 answers
899 views

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 ...
3 votes
1 answer
244 views

Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?)

Let $f:X\to S$ be a morphism between algebraic varieties which are smooth over a field of characteristic zero. We define the (derived) direct image functor $f_+:\mathsf{D}^b(\mathcal{D}_X)\to \mathsf{...
2 votes
0 answers
115 views

Local systems as a Serre subcategory of the category of perverse sheaves

Let $X$ be an algebraic variety. Let $Perv(X)$ be the (abelian) category of perverse sheaves on $X$ and let $Loc^{ft}(X)$ be the subcategory of local systems with finitely-generated stalks. It is ...
2 votes
0 answers
188 views

Intermediate extensions of pure perverse sheaves (BBD 5.4.3)

I am working my way through "Faisceaux pervers" by Beilinson, Bernstein and Deligne and can't wrap my head around Corollary 5.4.3. To me it seems that one of the hypotheses is extraneous, ...
8 votes
4 answers
3k views

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 $...
6 votes
1 answer
304 views

Understanding an involution of the category of perverse sheaves on $\mathbb{C}$

It is well-known (for example: chapter 2 in [GGM] A. Galligo, M. Granger, P. Maisonobe. D-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble, ...
28 votes
8 answers
4k views

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/...
4 votes
1 answer
354 views

Decomposition of direct image of a smooth morphism, Deligne's theorem, motives

Let $f : X \to Y$ be projective and smooth morphism of complex algebraic varieties. Here we care about the algebraic topology of $X$ and $Y$, so use classical topology for simplicity. I can take the ...
7 votes
0 answers
194 views

A reference for Bernstein's approach to KL conjectures

The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy. Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...
11 votes
0 answers
353 views

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 ...
4 votes
1 answer
431 views

Perverse sheaves on the complex affine line

Show that a perverse sheaf on $\mathbb{A}^1(\mathbb{C})$ (the complex plane with the analytic topology) is a bounded complex $A$ of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology ...
8 votes
3 answers
2k views

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-...
2 votes
1 answer
382 views

Purity of perverse cohomology sheaves

Let $f\colon X\to Y$ be a morphism of projective varieties over a finite field. Let $K$ be a perverse pure sheaf on $X$. Are the perverse cohomology sheaves of $f_*(K)$ pure? I am just learning the ...
3 votes
1 answer
504 views

Decomposition theorem over more general base schemes

The BBDG decomposition theorem says that if $f\colon X \to Y$ is a projective morphism of finite type $\mathbf{C}$-schemes and $X$ is smooth of (pure) dimension $d$ then $\mathbf{R}f_*\mathbf{Q}_\ell[...
6 votes
0 answers
258 views

The geometric "hands-on" vs. algebraic approach to nearby cycles

Feel free to skip to the question below; the following is just context and discussion: An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (...
2 votes
1 answer
518 views

A computation of intersection homology

I am reading about perverse sheaves from the notes of Cataldo and Migliorini http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf In page 553 example 2.2.2 they ...
2 votes
0 answers
201 views

Why does nearby cycles of a local system on $\mathbb{G}_m$ have same monodromy as local system?

Apologies if this belongs on MSE, but none of the tags I wanted existed, so I took it as a hint to post on MO. Edit: here is my definition of nearby cycles. Suppose $X$ is a complex analytic space ...
7 votes
0 answers
618 views

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 ...
6 votes
1 answer
448 views

About an application of BBD decomposition theorem

There is a following proposition in the famous paper on Koszul duality by Beilinson, Ginzburg and Soergel: let $G$ denote a semisimple complex Lie group, let $B$, $Q$ and $W^Q$ denote a pair of a ...