Questions tagged [intersection-cohomology]

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Hard Lefschetz theorem in intersection cohomology

In [1,2] the authors proved the Hard Lefschetz theorem in intersection cohomology: Let $Z$ be a complex projective variety of pure complex dimension $d$, with $\xi\in H^2(Z,\mathbb{Q})$ the first ...
3 votes
0 answers
124 views

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
294 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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4 votes
0 answers
290 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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3 votes
0 answers
137 views

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 $...
2 votes
1 answer
270 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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Anything similar to cone product formula (for convex polytopes)?

The convex polytope flag vector ring $\mathcal{R}$ satisfies the cone product formula $$ C(U) C(V) = C(J(U, V)) + DUV $$ where $$ J(U, V) = U C(V) + C(U) V - e_1 UV $$ is the join formula. Note: ...
6 votes
0 answers
134 views

Does there exist a notion of Chern classes in intersection cohomology?

First of all: I apologize for my mistakes, I'm a freshman in intersection cohomology. Let $X$ be a (compact) complex analytic space, let $L$ be a line bundle over $X$. Can one define a notion of ...
1 vote
0 answers
148 views

$\ell$-adic intersection cohomology of a nodal cubic over a finite field

I would like to know whether the $\ell$-adic intersection cohomology on the étale site of a projective nodal cubic over a finite field is trivial in degree $1$ (i.e., whether or not the first betti ...
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2 votes
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74 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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Kazhdan-Lusztig positivity of monomials in the Hecke algebra of a Coxeter System

In 1990, Deodhar [3] showed that the non-negativity of Kazdan-Lusztig polynomials implies the expansion of any monomial of (dual) Kazhdan-Lusztig basis elements $C'_{s_{i_1}}\cdots C'_{s_{i_r}}=\sum_x ...
4 votes
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IC sheaves and formal neighbourhoods

Let $X$ and $Y$ be two schemes of finite type over a finite field $\mathbb F_q$. Let $x$ (resp. $y$) be an $\mathbb F_q$-point of $X$ (resp. of $Y$). Let now $l$ be a prime which is prime to $q$. ...
2 votes
1 answer
367 views

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 ...
7 votes
0 answers
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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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1 vote
1 answer
245 views

What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure?

Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion ...
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6 votes
1 answer
402 views

Confusion about a proof from Goresky and MacPherson's "Intersection Homology II"

Context My question is about the "proof of claim" on page 84 of Goresky and MacPherson's "Intersection Homology II". For ease of reading, here's the claim: Claim: Suppose $X$ is a topological ...
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1 vote
1 answer
136 views

Internal product on intersection (co)homology

My question is short. Under what circumstances, if any, does there exist a well-defined internal product on the intersection cohomology groups of a pseudomanifold? I'm curious to know whether there is ...
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7 votes
1 answer
660 views

intersection cohomology and nearby cycles

This seems like a really basic question, but I somehow don't know and haven't been able to find the answer. I suspect that (at least under suitable assumptions) there should be a relation between ...
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9 votes
0 answers
186 views

Etale maps and local intersection cohomology

Suppose that $f:(X,x) \to (Y,y)$ is etale at $x$, meaning that it induces an isomorphism $C_xX \to C_yY$ on tangent cones. Then $f$ induces an isomorphism from the cohomology of $IC_{X,x}$ (the stalk ...
3 votes
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Where should I look for computing the intersection homology of projective varieties?

I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-...
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15 votes
1 answer
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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 ...
14 votes
2 answers
780 views

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 ...
14 votes
1 answer
707 views

Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?

In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ...
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7 votes
1 answer
627 views

A conjecture of Cheeger about intersection cohomology and $L^2$- cohomology

Let $X$ be a projective variety and let $D$ be a simple normal crossings divisor on $X$ Does $$IH^*(X;\mathbb C)\cong H_{(2)}^*(X\setminus D;\mathbb C)$$ hold true for each Kähler metric on $...
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6 votes
0 answers
485 views

What's the relationship between the different versions of the BBD decomposition theorem?

I have a few questions relating to the BBD decomposition theorem. I have come across the following two versions of the decomposition theorem. Version 1. Let $f : X \to Y$ be a proper map of ...
0 votes
0 answers
208 views

Determine existence of irreducible variety in given homology class

Given a homology class $\alpha \in H_k(X,\mathbb{Z})$ on a variety $X$, is there a way to determine if there exists an irreducible subvariety $Y \subset X$ that has that class, i.e. $[Y] = \alpha$? ...
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14 votes
2 answers
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Intersection Cohomology and $L^2$ cohomology

In the study of singular spaces, topological methods like intersection cohomology have played an important role. They have led to the development of technology like perverse sheaves and these find ...
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2 votes
2 answers
420 views

Example to show that the inverse image under a finite morphism is not t-exact with respect to the perverse t-structure

According to Chapter 4 of Beilinson, Bernstein, and Deligne's "Faisceaux Pervers" (Asterisque 100, 1980) the inverse image $Rf^*$ with respect to a finite morphism $f$ is right t-exact with respect to ...
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0 votes
0 answers
259 views

Homology class of variety defined by an ideal

if a subvariety of codimension n is given by an ideal of polynomials with n generators, then the homology class of the variety is given by the intersection product of the classes of the individual ...
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7 votes
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Local intersection cohomology

Let $X$ be a variety and $p\in X$ a point. Let $IC_X$ be the intersection cohomology sheaf, and let $IC_{X,p}$ be its stalk at $p$. Let $IH^*_p(X) := H^{*-\dim X}(IC_{X,p})$ be the local ...
3 votes
1 answer
368 views

intersection complex for quotient singularities

Let $X$ be a projective variety over a field of characteristic zero and assume that $X$ has finite quotient singularities, that is, $X$ is a union of affine open subsets of the form $Y/G$, where $G$ ...
3 votes
0 answers
682 views

Canonical basis of quantum groups

I am trying to understand the canonical basis of quantum groups and different ways to construct the canonical basis of quantum groups. In the comments of Lusztig's papers, the paper [92], CANONICAL ...
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1 vote
2 answers
240 views

on a characterisation of the intersection complex

Let $X$ be an integral scheme of finite type over a field $k$ of dimension $d$ and $U$ an open dense smooth subscheme. Let $K\in D_{c}^{b}(X,\bar{\mathbb{Q}}_{l})$ be such that $K_{U}=\bar{\mathbb{Q}}...
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6 votes
0 answers
180 views

Divisibility of all entries in an intersection form

What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer? More precisely: $F \subset S$ is a proper connected (usually reducible) half-...
9 votes
3 answers
1k views

Applications for intersection (co)homology and for the Decomposition Theorem for students?

Which applications of intersection (co)homology and of the (Topological) Decomposition Theorem have most chances to be understood by students?
5 votes
0 answers
615 views

intersection cohomology and etale cohomology

Hello, Can someone explain or give a reference on the comparison between intersection cohomology and l-adic etale cohomology of a variety over a field of characteristic zero? Thanks!
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16 votes
1 answer
583 views

Is there a notion of a chain complex with corners?

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes. On the topological side, although in the PL ...
3 votes
2 answers
740 views

Non-vanishing of cup product in cohomology

Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $\alpha \in H^{2k}(X)$ be an algebraic cohomology class. Let us fix an $l$ such that $H^l(X) \neq 0$ and $H^{l+2k}(X) \neq 0$. The ...
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3 votes
1 answer
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Bounding the size of stalks of IC sheaves

Say $X$ is a smooth algebraic variety, $U$ is a Zariski open set in $X$, $L$ is a local system on $U$, and $IC(L)$ the intersection cohomology sheaf on $X$ which restricts to $L$ on $U$. Then is: $$...
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7 votes
1 answer
564 views

Does intersection pairing on `$IH^*(X)$` agree with cup-product on `$H^*(X)$`?

Let $X$ be a proper singular variety over $k=\overline{\mathbb F}_p,$ irreducible of dimension $d.$ Let $H^*(X)$ and $IH^*(X)$ be the $l$-adic cohomology groups and $l$-adic intersection cohomology ...
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6 votes
2 answers
961 views

Stratified pseudomanifold

In the definition of an $n$-dimensional stratified pseudomanifold one demands the following filtration $X=X_n \supset X_{n-1}=X_{n-2} \supset X_{n-3}\supset ... \supset X_0 \supset X_{-1}=\emptyset$. ...
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2 votes
0 answers
395 views

The signature of a mapping torus

Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold $B$...
10 votes
1 answer
493 views

"geometric" interpretation of the alternating sum of intersection cohomology groups

Let $X_0$ be a proper variety over a finite field $k.$ For each prime number $\ell\ne p,$ we have the $\ell$-adic intersection cohomology groups $IH^i(X).$ Due to Gabber, the alternating sum of these ...
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4 votes
1 answer
716 views

intersection pairing on intersection cohomology

Let $X$ be a projective variety of dimension $d$ over $k=\bar{k},$ with $L$ an ample line bundle on $X$ and $\eta=c_1(L).$ Hard Lefschetz gives an isomorphism (see BBD) $$ \eta^i:IH^{d-i}(X)\to IH^{d+...
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7 votes
0 answers
861 views

Poincaré duality for intersection cohomology

Let $X$ be a projective complex algebraic variety of dimension $d.$ Does it make sense to ask if properties like $$ (x,y)=(-1)^i(y,x) $$ holds, for $x\in IH^i(X,\mathbb Q)$ and $y\in IH^{2d-i}(X)?$ ...
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6 votes
1 answer
580 views

Intersection Cohomology of Coordinate Hyperplanes

I'm trying to learn how to compute stalks of IC sheaves, and I was wondering about the following example: Fix $n$. Let $X \subset \mathbb{C}^n$ be the variety cut out by the equation $x_1 \cdots x_n =...