Questions tagged [perverse-sheaves]

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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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567 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 (...
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1answer
283 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 ...
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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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1answer
231 views

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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111 views

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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243 views

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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134 views

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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1answer
185 views

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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60 views

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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2answers
373 views

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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813 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-...
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107 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 ...
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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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50 views

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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1answer
140 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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1answer
154 views

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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1answer
69 views

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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152 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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1answer
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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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1answer
441 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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1answer
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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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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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807 views

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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1answer
177 views

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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1answer
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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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1answer
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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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119 views

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

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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1answer
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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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2answers
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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 ...