# Questions tagged [perverse-sheaves]

The perverse-sheaves tag has no usage guidance.

179
questions

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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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### 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 ...

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397
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### 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 ...

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### 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 $*$ ...

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186
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### 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 ...

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### $K$-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{...

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### Hodge structure on intersection cohomology of toric varieties

Given a convex polytope with integer vertices, one can construct a complex projective variety $X$ called toric variety. In general $X$ is not smooth. As I have heard, by the work of M. Saito, the ...

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### Hodge theoretic properties of intersection cohomology

Let $X$ be a complex projective irreducible reduced variety. It is well known that the intersection cohomology of $X$ satisfies versions of Poincare duality and hard Lefschetz theorem.
Does it admit a ...

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### Perverse sheaves and maximal genus Gopakumar-Vafa invariants

Let $f: X \to Y$ be a proper morphism between complex varieties (the varieties as well as the map may be non-smooth) and let $\phi \in \text{Perv}(X)$ be a perverse sheaf on $X$. Given this data, it ...

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### Absolute purity for intersection cohomology

If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then
$$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$
where $(\star)$ is true when $i$ is in addition regular.
Here ...

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### Perverse restriction

Let $f:U\to V$ be a separable dominant morphism of irreducible positive-dimensional varieties. Let $F$ be a perverse sheaf on $U$. Are there infinitely many closed points $p\in V$ such that $F|_{U_p}$ ...

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### Intersection homology of toric resolutions

I'm interested in the intersection homology of toric varieties associated to a polytope $P$ with proper faces F, and a subdivision $P'$ of P. Let $X_P$ be the toric variety associated to the polytope $...

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### intuition about perverse sheaves

firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...

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### Who introduced the heart ($\mathcal{C}^\heartsuit$) notation in the context of $t$-structures on triangulated categories?

In the context of $t$-structures
([Wikipedia],
[nLab],
[Notes I],
[Notes II],
[HA, Definition 1.2.1.11)],
[BBD, Définition 1.3.1]),
one often writes $\mathcal{C}^\heartsuit$ for the heart of a ...

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### Example of an intersection complex not concentrated in a single degree

I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful.
I want to construct an example of an intersection ...

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### pointwise purity for character sheaves on a wonderful compactification

Consider the minimal (Goresky-MacPherson) extension of a character sheaf on a semi-simple (say) adjoint group to its wonderful compactification. Is that extension pointwise pure?

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### is constant sheaf perverse on a Cohen-Macaulay variety?

Let $X$ be a connected Cohen-Macaulay algebraic variety of dimension $d$, say, over $\mathbb C$. Is it true that $\underline{{\mathbb C}_X}[d]$ is a perverse sheaf, where $\underline{{\mathbb C}_X}$ ...

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### Singular support of an irreducible perverse sheaf

I was studying Sheaves on Manifolds by Kashiwara and Schapira, and while the singular support seems like a complicated invariant I cannot seem to find a counterexample to the following:
Let $X$ be a ...

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### 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-...

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### 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 ...

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### Stalks of perverse cohomology sheaves?

For a complex of sheaves $\cal{F}^{\bullet}$ on a variety $X$, a useful fact is that the stalks of the cohomology sheaves of $\mathcal{F}^{\bullet}$ agree with the cohomology groups of the complex of ...

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### Applications of character sheaves

There are many important recent works (for example, by Lusztig, Bezrukavnikov-Finkelberg-Ostrik, Ben-Zvi-Nadler, Boyarchenko-Drinfeld, Lusztig-Yun, Vilonen-Xue) on character sheaves (which are certain ...

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### So what exactly are perverse sheaves anyway?

Is there a way to define perverse sheaves categorically/geometrically? Definitions like the following from lectures by Sophie Morel:
The category of perverse sheaves on $X$ is $\mathrm{Perv}(X,F):=...

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### Is the characteristic cycle map for perverse sheaves injective?

Let $X$ be a smooth irreducible complex variety. Is the characteristic cycle map from the Grothendieck group of perverse sheaves (with complex coefficients) on $X$ to the free abelian group generated ...