# Questions tagged [intersection-cohomology]

The tag has no usage guidance.

50 questions
Filter by
Sorted by
Tagged with
480 views

### Intersection cohomology and Poincaré duality

When trying to learn about perverse sheaves I hand-wavingly thought that intersection cohomology is the ‘minimal’ way of fixing the failure of Poincaré duality. But I am very aware that it is risky to ...
113 views

### Comparison of IC sheaves on Schubert varieties on two settings (l-adic vs. complex)

This question is basically about comparison of IC sheaves (or their sheaf cohomologies) for the settings: 1. variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear, 2. variety is over a finite ...
100 views

### Triple insersection number of a surface in three-manifolds

I heard something about the triple intersection number $\text{mod}(2)$ (but maybe also $\text{mod}(n)$) of a surface in an orientable three-manifold but I couldn't find a precise definition. My guess ...
205 views

405 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 ...
93 views

### 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: ...
171 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
169 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 ...
87 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 ...
147 views

1 vote
273 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 ...
432 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 ...
1 vote
146 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 ...
773 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 ...
195 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 ...
390 views

### 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-...
527 views

### 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 ...
874 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 ...
774 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 ...
653 views

185 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-...
2k 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?
670 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!
598 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 ...
766 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 ...
351 views

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: $$... 7 votes 1 answer 585 views ### Does intersection pairing on IH^*(X) agree with cup-product on H^*(X)? Let X be a proper singular variety over k=\overline{\mathbb F}_p, irreducible of dimension d. Let H^*(X) and IH^*(X) be the l-adic cohomology groups and l-adic intersection cohomology ... 6 votes 2 answers 1k views ### Stratified pseudomanifold In the definition of an n-dimensional stratified pseudomanifold one demands the following filtration X=X_n \supset X_{n-1}=X_{n-2} \supset X_{n-3}\supset ... \supset X_0 \supset X_{-1}=\emptyset. ... 2 votes 0 answers 418 views ### The signature of a mapping torus Consider a manifold M of dimension 4k + 2, k an integer. Pick a diffeomorphism \phi of M and construct the mapping torus T of \phi. Suppose that there is a 4k+4 dimensional manifold B... 10 votes 1 answer 519 views ### "geometric" interpretation of the alternating sum of intersection cohomology groups Let X_0 be a proper variety over a finite field k. For each prime number \ell\ne p, we have the \ell-adic intersection cohomology groups IH^i(X). Due to Gabber, the alternating sum of these ... 4 votes 1 answer 755 views ### intersection pairing on intersection cohomology Let X be a projective variety of dimension d over k=\bar{k}, with L an ample line bundle on X and \eta=c_1(L). Hard Lefschetz gives an isomorphism (see BBD)$$ \eta^i:IH^{d-i}(X)\to IH^{d+...
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 $y\in IH^{2d-i}(X)?$ ...
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 =...