# 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 ...
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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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### 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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### 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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### 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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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 $^PH^0(\mathcal{F} \boxtimes^... • 637 1 vote 1 answer 245 views ### What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure? Let$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$denote the inclusion ... • 155 6 votes 1 answer 402 views ### Confusion about a proof from Goresky and MacPherson's "Intersection Homology II" Context My question is about the "proof of claim" on page 84 of Goresky and MacPherson's "Intersection Homology II". For ease of reading, here's the claim: Claim: Suppose$X$is a topological ... • 2,981 1 vote 1 answer 136 views ### Internal product on intersection (co)homology My question is short. Under what circumstances, if any, does there exist a well-defined internal product on the intersection cohomology groups of a pseudomanifold? I'm curious to know whether there is ... • 35 7 votes 1 answer 660 views ### intersection cohomology and nearby cycles This seems like a really basic question, but I somehow don't know and haven't been able to find the answer. I suspect that (at least under suitable assumptions) there should be a relation between ... • 2,679 9 votes 0 answers 186 views ### Etale maps and local intersection cohomology Suppose that$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}$(the stalk ... • 2,343 3 votes 0 answers 344 views ### Where should I look for computing the intersection homology of projective varieties? I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-... • 1,656 15 votes 1 answer 494 views ### IC sheaf of certain explicit variety Let$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$. QUESTION: What is the stalk (with the action ... • 6,677 14 votes 2 answers 780 views ### Non semi-simple monodromy in an algebraic family I am looking for an example of a (edit: projective) family$f : X \to Y$of complex algebraic varieties which is a topologically locally trivial fibration in (singular) varieties and such that there ... • 5,567 14 votes 1 answer 707 views ### Why should intersection cohomology and quantum cohomology be related for a symplectic resolution? In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ... • 2,199 7 votes 1 answer 627 views ### A conjecture of Cheeger about intersection cohomology and$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$... 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 ...
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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 $G$ ...
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### 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 , CANONICAL ...
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