# Questions tagged [perverse-sheaves]

The perverse-sheaves tag has no usage guidance.

139
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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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138 views

### 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 ...

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116 views

### Intermediate extension and irreducible subquotients of perverse cohomologies

Let a set X be the union of two locally closed subsets U and V such that U does not lie in the closure of V. Let the restriction of a complex R of constructible sheaves on X to a smooth open subset A ...

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### Calculating intermediate extension on the stack of coherent sheaves of rank $1$

Let $L$ be a line bundle of degree $d$ on a curve $X$ and let $x$ be a point of $X$. I want to describe the intermediate extension of the constant sheaf from the stack of line bundles of degree $d$ to ...

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### Intermediate extension and perverse cohomologies

Let a set X be the union of two locally closed subsets U and V such that U does not lie in the closure of V. Let the restriction of a complex R of constructible sheaves on X to a smooth open subset A ...

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### Open restriction and Fourier transform on irreducible subquotients of perverse cohomologies

Is it true that an open restriction to $U \subset X$ induces a surjection on the set of irreducible perverse subquotients of perverse cohomologies (i.e. cohomologies with respect to the perverse t-...

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141 views

### Nearby cycles and tensor product

So suppose we have a projective smooth map $f: X \rightarrow A^1$. Then we have a nearby cycles functor $\psi_f$ that sends constructible complexes on $X$ to complexes on $X_1$. In general, I think ...

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### Does intermediate extension functor commutes with forgetful functor in equivariant derived category?

The forgetful functor from $D^b_G(X)$ to $D^b(X)$ carries $Perv_G(X)$ to $Perv(X)$ by definition $5.1$ in the book of Bernstein and Lunts. My question is do the following functors, intermediate ...

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### Riemann Hilbert Correspondence with fixed stractification

Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category ...

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### Espace étalé for derived category

It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{...

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### 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 ...

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### Explicit computation for perverse cohomology

To construct the convolution product for two ($G(O)$-equivariant) perverse sheaves $\mathcal{F}, \mathcal{G}$ on affine grassmanian, the first thing we need to compute is $^PH^0(\mathcal{F} \boxtimes^...

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### Families of Hessenberg varieties for $GL_n$

In short, the question is
What do we know about the sheaf $\pi_*\underline{\bar{\mathbb{Q}}_{\ell}}$ given by the family of (very original, see below) Hessenberg varieties for $GL_n$? As a sum of ...

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359 views

### Fulton's deformation to the normal cone vs Verdier's

Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone:
Verdier's version: $\tilde{X}_Y^\...

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### Additivity of characteristic cycle of holonomic D-module

Let $\mathcal{M}$ be a holonomic D-module on a complex analytic (or alternatively, algebraic) manifold $X$. One can attach to it (using a good filtration) a characteristic cycle $Ch(\mathcal{M})$ ...

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323 views

### Lefschetz pencils and perverse sheaves

I have been reading BBD and Geordie Williamson's “An illustrated guide to perverse sheaves”. The latter has been tremendously helpful to get some intuition for the former.
Let $K$ be some field, and ...

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### Stability of mixed complexes under open embeddings

In Weil II, Deligne proves that the six functors preserve the category of mixed $l$-adic complexes. Most difficulties are encountered when proving that if $f\colon X\to Y$ is a finite type separated ...

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### consequence of the definition of perverse sheaves

I am trying to learn perverse sheaves. They are complexes $M$ of sheaves with constructible cohomology (say we are working with algebraic varieties and the transcendental topology) such that the ...

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### Counterexamples to gluing complexes of sheaves

Note: I asked the question below last week on MathSE but received no answer.
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This ...

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### Convolution of $\ell$-adic sheaves and group homomorphisms

This question follows this one , where I defined convolution of $\ell$-adic/perverse sheaves.
Here I am working with a perfect field $k$ ($char(k)\neq l$) and with a smooth separated groupscheme $G$ ...

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### Convolution of $\ell$-adic sheaves is commutative if the group is commutative

[This is a duplicate of this question on Stackexchange]
I am trying to figure out how to prove a very basic statement about convolution of $\ell$-adic/perverse sheaves in Katz's "Rigid local systems" ...

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### Computation of multiplicity of irreducible representation in some representation via geometric Satake correspondence

Let $G$ be a reductive algebraic group over $\mathbb{C}$. Let $\operatorname{Gr}_{G}$ be the corresponding affine Grassmannian ($\operatorname{Gr}_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$...

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### A property of nearby cycles functor

Let $f\colon X\to Y$ be a flat morphism of irreducible projective algebraic varieties over $\mathbb{C}$ (or any other algebraically closed field of characteristic 0). Assume that $Y$ is smooth, and ...

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### Base change and the octahedron axiom

I am trying to understand "de Cataldo, Migliorini. The perverse filtration and the Lefschetz hyperplane theorem. Annals of Mathematics, 171(2010), 2089-2113." My question is about one detail in the ...

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### $\mathrm{Hom}_D(D^{\geq 1},D^{\leq 0})=0?$

Let $D$ be a triangulated category with $t$-structure given by strictly full subcategories $D^{\leq0},D^{\geq 1}$. By definition we have $\mathrm{Hom}_D(D^{\leq 0},D^{\geq 1})=0$.
Can we decuce from ...

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### Nearby cycle functor for a family of stable curves

Let $B$ be a smooth algebraic curve over $\mathbb{C}$ (or rather a germ of it at a point $b\in B$). Let $f\colon E\to B$ be a proper flat family of stable curves with smooth generic fiber. Assume that ...

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### 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/...

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### “Correct” definition of stratified spaces and reference for constructible sheaves?

It seems that the theory of constructible sheaves (in particular anything that goes into proving that they form an abelian category) requires some technical statements about existence of certain ...

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### On the definition of triangulated categories

Triangulated categories were introduced in the 1960s by Grothendieck and Verdier in order to develop homological algebra in the framework of derived categories. An example of a triangulated category ...

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### Springer fibers and Weyl group

Let $\pi:\tilde{\mathfrak{g}}\rightarrow\mathfrak{g}$ the Grothendieck-Springer resolution of a semisimple Lie algebra $\mathfrak{g}$, over $\mathbb{C}$.
We know it's a small map, and that $\pi_{*}\...

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### When is $\mathbb{Q}_X$ pure?

I'll ask this question in the language of mixed Hodge modules, since that's where I'm coming from, but the question has an exact analogue for mixed l-adic complexes on schemes over fields of positive ...

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### on the Springer sheaf

Let $\pi:\tilde{\mathfrak{g}}\rightarrow \mathfrak{g}$, be the Grothendieck-Springer resolution, where $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$.
We know that $\pi$ is small thus $\...

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### On the notion of conelike stratified (cs-) space

The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's ...

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### Relation between crystalline and perverse sheaves

Take $X$ to be a smooth complex projective algebraic variety. The Riemann-Hilbert correspondence gives an equivalence of categories between the category of perverse sheaves on $X$ and the category of ...

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### derived invariants, perversity and modular coefficients

Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$.
Let $n$ an integer such that it is not prime with the order of $\Gamma$.
Then $\pi_{*}\mathbb{Z}/n\...

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### IC sheaf of certain explicit variety

Let $n,m$ be two positive integers. Let $Z$ denote the closed subvariety
in $\mathbb A^n \times \mathbb A^m$
given by the equation $x_1...x_n=y_1...y_m$.
QUESTION: What is the stalk (with the action ...

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### functions coming from a perverse sheaf

Let's take a scheme $X$ over a finite field $k$ and $f:X(k)\rightarrow\mathbb{Q}_{\ell}$
What kind of condition do I need on $f$ if I want that it comes from an irreducible perverse sheaf on $X$?

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### What is an example of a non-mixed $\ell$-adic sheaf?

$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...

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### Perverse sheaves and tensor product

If $X$ is a connected algebraic variety of finite type over $k$ (with $k$ a field of positive characteristic) of dimension $d$, and if $\mathcal{F}$ and $\mathcal{G}$ are perverse sheaves on $X$ so $(\...

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### Examples of calculating perverse sheaves on algebraic varieties with easy stratification

I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
http://www.amazon.com/Introduction-...

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### Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors
$$
...

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### Non semi-simple monodromy in an algebraic family

I am looking for an example of a (edit: projective) family
$f : X \to Y$
of complex algebraic varieties which is a topologically locally trivial fibration in (singular) varieties and such that there ...

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### Gabber's original proof of his purity theorem

Gabber's purity theorem is the statement that if $\mathscr{F}$ is a pure perverse sheaf on an open subvariety $j : U \hookrightarrow X$ then so is $j_{!*} \mathscr{F}$.
It is remarkable because it ...