Questions tagged [perverse-sheaves]

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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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7 votes
0 answers
176 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-...
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9 votes
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266 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 ...
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4 votes
1 answer
321 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 ...
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2 votes
1 answer
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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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3 votes
1 answer
333 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[...
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6 votes
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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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4 votes
1 answer
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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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2 votes
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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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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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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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1 answer
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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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26 votes
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693 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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5 votes
1 answer
341 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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3 votes
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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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4 votes
1 answer
290 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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2 votes
0 answers
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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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4 votes
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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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4 votes
0 answers
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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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3 votes
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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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10 votes
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516 views

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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3 votes
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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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2 votes
1 answer
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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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2 votes
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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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6 votes
0 answers
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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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3 votes
2 answers
486 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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8 votes
3 answers
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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-...
3 votes
1 answer
174 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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3 votes
0 answers
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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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14 votes
0 answers
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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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19 votes
3 answers
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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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3 votes
0 answers
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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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2 votes
1 answer
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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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2 votes
0 answers
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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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  • 2,189
2 votes
1 answer
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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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  • 2,189
3 votes
1 answer
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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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  • 2,189
2 votes
0 answers
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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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2 votes
0 answers
73 views

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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5 votes
1 answer
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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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1 vote
0 answers
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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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2 votes
1 answer
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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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7 votes
0 answers
144 views

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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4 votes
0 answers
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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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6 votes
1 answer
521 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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6 votes
1 answer
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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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8 votes
0 answers
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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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