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Questions tagged [perverse-sheaves]

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What advantages do perverse sheaves provide over D-modules? (or vice versa)

My question is as in the title: What advantages do perverse sheaves provide over D-modules? (or vice versa) As a specific example: could something like the modular generalized Springer correspondence ...
Andrea B.'s user avatar
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Decomposition theorem for resolution of surface singularity

I’ve asked this question on MSE but I don’t get an answer, so I’m trying to ask here. https://math.stackexchange.com/questions/4914142/decomposition-theorem-for-resolution-of-surface-singularities In ...
TaiatLyu's user avatar
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3 votes
1 answer
130 views

Is inverse image along finite group quotient $t$-exact for the perverse $t$-structure?

Let $q: \mathbb{C}\to \mathbb{C}$ be the quotient by $\mathbb{Z}/n\mathbb{Z},$ i.e. the map taking $z\mapsto z^n$. In the accepter answer to Operations on perverse sheaves on disk the inverse image of ...
Sergey Guminov's user avatar
2 votes
1 answer
150 views

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 ...
arczn's user avatar
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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 ...
Mathstudent's user avatar
4 votes
0 answers
143 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 ...
Sergey Guminov's user avatar
4 votes
1 answer
207 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 ...
Tanny Sieben's user avatar
3 votes
0 answers
149 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 ...
Sergey Guminov's user avatar
5 votes
1 answer
568 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 ...
l-r-b's user avatar
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168 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 ...
Pulcinella's user avatar
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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 ...
Doug Liu's user avatar
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101 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?
user141099's user avatar
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52 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 ...
BnPrs's user avatar
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1 answer
336 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). ...
Doug Liu's user avatar
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0 answers
129 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 ...
Ji Woong Park's user avatar
7 votes
1 answer
440 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 ...
Antoine Labelle's user avatar
2 votes
0 answers
119 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 ...
Yellow Pig's user avatar
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8 votes
0 answers
329 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 ...
Sergey Guminov's user avatar
2 votes
1 answer
337 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 $k=\overline{k}$, $D_i, i=1,...,r$ ...
Allen Lee's user avatar
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2 answers
252 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 $...
JackYo's user avatar
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0 answers
183 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 ...
Martin Tang's user avatar
1 vote
1 answer
170 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(...
Sergey Guminov's user avatar
3 votes
0 answers
332 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 $\...
zygomatic's user avatar
4 votes
1 answer
134 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 ...
Sergey Guminov's user avatar
2 votes
0 answers
186 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) ...
Tommaso Scognamiglio's user avatar
4 votes
0 answers
325 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 ...
Michael Barz's user avatar
3 votes
1 answer
319 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 $\...
random123's user avatar
  • 411
2 votes
0 answers
86 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{...
Doug Liu's user avatar
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2 votes
0 answers
78 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 ...
W. Rether's user avatar
  • 435
3 votes
1 answer
257 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{...
Gabriel's user avatar
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2 votes
0 answers
124 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 ...
Laurent Cote's user avatar
2 votes
0 answers
195 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, ...
Sergey Guminov's user avatar
6 votes
1 answer
310 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, ...
chan kifung's user avatar
7 votes
0 answers
198 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-...
Rgdn Dznrbx's user avatar
11 votes
0 answers
367 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 ...
Tommaso Scognamiglio's user avatar
4 votes
1 answer
436 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 ...
Stabilo's user avatar
  • 1,479
2 votes
1 answer
397 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 ...
asv's user avatar
  • 21.3k
3 votes
1 answer
538 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[...
gdb's user avatar
  • 2,873
6 votes
0 answers
260 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 (...
Mathmank's user avatar
  • 273
4 votes
1 answer
382 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 ...
Geordie Williamson's user avatar
2 votes
0 answers
223 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 ...
Mathmank's user avatar
  • 273
8 votes
0 answers
667 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 ...
EJAS's user avatar
  • 191
6 votes
1 answer
462 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 ...
John S.'s user avatar
  • 111
1 vote
0 answers
161 views

When does a $D$-module think it’s a pullback along a smooth morphism?

Let $X$ and $Y$ be two algebraic varieties, and let $f: X \to Y$ be a morphism. Suppose $A$ is a holonomic $D$-module on $Y$. In this situation we can pull $A$ back to $X$ using either the $!$ or $*$ ...
Exit path's user avatar
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1 vote
0 answers
215 views

On hypercohomology of perverse sheaves

I was watching this Youtube video on a lecture given by J.P. Brasselet on perverse sheaves. At around 54:37 he mentioned the following result: Let $X=\cup X_\alpha$ be a Whitney stratified space of ...
Winnie_XP's user avatar
  • 287
5 votes
1 answer
415 views

Tensor product of perverse sheaves on flag varieties

I am interested in computing tensor products of perverse sheaves on (partial) flag varieties. For a specific example - consider the product of the big projective on $\mathbb{P}^1$ with itself (This is ...
Adam Gal's user avatar
  • 690
4 votes
0 answers
143 views

Bialgebras from mixed Bruhat sheaves

Let $A = \bigoplus_{i=0}^{+\infty} A_i$ be a graded bialgebra in a braided monoidal category $\mathcal V$. Then, according to Kapranov–Schechtman's article "Parabolic induction and perverse ...
Nicolas Hemelsoet's user avatar
27 votes
0 answers
934 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 (...
Geordie Williamson's user avatar
5 votes
1 answer
486 views

Are equivariant perverse sheaves constructible with respect to the orbit stratification?

[Moved here from MSE] Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$. Question. Is it true ...
W. Rether's user avatar
  • 435
8 votes
0 answers
422 views

$K$-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{...
user127776's user avatar
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