Questions tagged [perverse-sheaves]
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206 questions
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Tate conjecture for singular varieties in terms of intersection homology
In his book “Mixed motives and algebraic K-theory”, Jannsen generalizes the Tate conjecture to a potentially singular projective variety $X$ over a finitely generated field. The statement is the same ...
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Essential image of nearby cycles functor
Let $R$ be a Henselian discrete valuation ring. Let $S=\mathrm{Spec} R$ be the corresponding trait, with generic point $\eta$ and closed point $s$. Let $f:X\to S$ be a smooth proper morphism of ...
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Is there an "$\ell$-adic Riemann Hilbert correspondence"?
The Riemann-Hilbert correspondence (see, e.g., Thm. 7.2.2 of D-modules, perverse sheaves, and representation theory) shows that analytic perverse sheaves are equivalent to regular holonomic $D$-...
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About decomposition theorem BBD with respect to some stratification
I want to follow up a question from here (how to deduce version 1.a. from version 1).
I know a version of decomposition theorem BBD:
Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex ...
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Cohomological range of a perverse sheaf
I want to prove that, let $X$ be a smooth variety, then for a local system $L$ on $X$, $L[\dim X]$ has no quotient object in the abelian category of perverse sheaves supported on a proper closed ...
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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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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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Monodromic but not equivariant sheaves and Braden's theorem
Let $X$ be a complex variety with contracting $\mathbb{G}_m$ action. Let $i\colon \{x_0\}\to X$ be the inclusion of the fixed point. Then the simplest case of Braden's theorem (which would then seem ...
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Fourier transform for perverse sheaves
I am interested in studying the Fourier transform for perverse sheaves on "nice" spaces, say affine complex space stratified by the action of an algebraic group into finitely many orbits.
In ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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$\pm 1$-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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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). ...
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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 ...
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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 ...
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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 ...
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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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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 $k=\overline{k}$, $D_i, i=1,...,r$ ...
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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 $...
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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 ...
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$\text{Ext}$-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(...
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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 $\...
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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 ...
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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) ...
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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 ...
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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 $\...
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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{...
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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 ...
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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{...
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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 ...
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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, ...
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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,
...
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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-...
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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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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 ...
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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 ...
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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[...
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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 (...
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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 ...
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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 ...
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