Intersection Cohomology of Coordinate Hyperplanes

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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 $x_1 \cdots x_n =0$, i.e. the coordinate hyperplanes. What are the stalks of $\mathrm{IC}(X)$ at the various points of $X$, in particular at the origin?

This seems like a natural toy example, but if the general answer is difficult, I'd be happy to know how to compute this for small $n$.

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1 Answer

Let $Y$ denote the disjoint union of the coordinate hyperplanes in $\mathbb{C}^n,$ and let $f:Y \to X$ denote the corresponding resolution of singularities.

1) Show that $f_{\ast}\mathbb{C}_Y[n-1] \simeq IC_X$ (consider, for example, the support conditions and the fact that both sheaves are isomorphic to $\mathbb{C}_U[n-1]$ when restricted to the nonsingular open $U \subset X$).

Edit (some details added): Letting $U$ denote the complement of the set where any two coordinate planes intersect, $f$ is an isomorphism when restricted to $U.$ We therefore have that the restriction (i.e., pullback) of $f_{\ast}\mathbb{C}_Y[n-1]$ to $U$ coincides with $\mathbb{C}_U[n-1]$ (by proper base change if you like).

In order to conclude that $f_{\ast}\mathbb{C}_Y[n-1] \simeq IC_X,$ we now just need to check the support and cosupport conditions which uniquely define the intersection cohomology sheaf (together with the fact that its restriction to $U$ is the (shifted) constant sheaf). These conditions are similar to, but more restrictive than, the support and cosupport conditions for perverse sheaves.

I recommend looking at page 21 of the wonderful article by de Cataldo and Migliorini, which can be found at http://arxiv.org/abs/0712.0349 for a statement of these support and cosupport conditions (and figure 1 on page 25 for a visual illustration of the definition).

Since the fibers of $f$ consist of a finite number of points, the cohomology of the fibers is non-zero only in degree zero. This shows that the first condition (the support condition) is satisfied.

For the second condition (the cosupport condition), you can either derive it from the support condition using Verdier duality and the properness of $f,$ or you can simply note that an open ball in $\mathbb{C}^{n-1}$ has non-zero compactly supported cohomology only in degree $2n-2.$

2) Now it's straightforward to compute any of the stalks since the fiber of $x \in X$ consists of anywhere between one point and n points, depending on how many hyperplanes $x$ lives inside of.

Alternatively, it is also possible to do this by using only basic definitions. To compute the stalk at $x,$ intersect a sufficiently small open ball around $x$ in $\mathbb{C}^n$ with $X$ and then calculate the intersection cohomology by considering intersection cochains (just like you would for singular cohomology, but now with a less restrictive notion of cochain).

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Can you elaborate on the argument in part 1? I'm sorry to belabor it, but I think I need it spelled out to me. – Dinakar Muthiah Nov 27 2010 at 18:35
A small detail, which makes me nervous: In "D-modules, perverse sheaves and Representation theory", they don't even define IC-complexes on non irreducible varieties. Given a not necessarily irreducible variety with equidimensional components. Is there still a unique complex which coincides on a dense open subset with a given shifted local system, and satisfies the (co)support condition for IC complexes? – Jan Weidner Nov 28 2010 at 8:36

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Jan, that's certainly a good point to bring up. I think they assume irreducible throughout because it makes things less messy and I think it's necessary for the decomposition theorem. However, if you look at Goresky and MacPherson's "Intersection Homology II," you'll see that the IC sheaf can be defined for a fairly large class of topological spaces known as pseudomanifolds (maybe you knew this already) and there is still a uniqueness result like the one you ask for (see 4.1 and 6.1 of "Intersection Homology II"). In the OP's question, $X$ is pure dimensional which means we should be ok. – Mike Skirvin Nov 28 2010 at 16:36

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One further comment: On an irreducible variety, IC sheaves are indecomposable in the derived category, as stated in Corollary 2 of section 4.1 of "Intersection Homology II." This would certainly seem to be a good reason for usually dealing with irreducible varieties. – Mike Skirvin Nov 28 2010 at 16:44
Okay, thanks very much Mike! – Jan Weidner Nov 28 2010 at 20:48
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