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3
votes
0answers
118 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
0answers
154 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. $[Y] = \alpha$? ...
8
votes
2answers
200 views

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 ...
1
vote
2answers
142 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 ...
0
votes
0answers
76 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 ...
4
votes
0answers
127 views

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 ...
2
votes
1answer
139 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 $G$ ...
2
votes
0answers
205 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 ...
1
vote
2answers
191 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 ...
5
votes
0answers
136 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) ...
7
votes
3answers
849 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?
3
votes
0answers
286 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!
16
votes
1answer
499 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 ...
2
votes
2answers
544 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 ...
2
votes
1answer
251 views

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: ...
6
votes
1answer
434 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 ...
3
votes
2answers
470 views

Stratified Pseudomanifold

Hi there, I have a, I guess, simple question. 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 ...
2
votes
0answers
267 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 ...
8
votes
1answer
374 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 ...
3
votes
1answer
577 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 ...
7
votes
0answers
651 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 $y\in IH^{2d-i}(X)?$ ...
4
votes
1answer
421 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 ...