# Questions tagged [intersection-cohomology]

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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 $q$. ...
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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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 $G$ ...