Intersection Cohomology of Coordinate Hyperplanes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:35:13Z http://mathoverflow.net/feeds/question/47504 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47504/intersection-cohomology-of-coordinate-hyperplanes Intersection Cohomology of Coordinate Hyperplanes Dinakar Muthiah 2010-11-27T13:06:48Z 2010-11-28T06:33:30Z <p>I'm trying to learn how to compute stalks of IC sheaves, and I was wondering about the following example:</p> <p>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?</p> <p>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 $n$.</p> http://mathoverflow.net/questions/47504/intersection-cohomology-of-coordinate-hyperplanes/47519#47519 Answer by Mike Skirvin for Intersection Cohomology of Coordinate Hyperplanes Mike Skirvin 2010-11-27T17:55:11Z 2010-11-28T06:33:30Z <p>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.</p> <p>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 $U \subset X$).</p> <p>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).</p> <p>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.</p> <p>I recommend looking at page 21 of the wonderful article by de Cataldo and Migliorini, which can be found at <a href="http://arxiv.org/abs/0712.0349" rel="nofollow">http://arxiv.org/abs/0712.0349</a> for a statement of these support and cosupport conditions (and figure 1 on page 25 for a visual illustration of the definition).</p> <p>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.</p> <p>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 $2n-2.$</p> <p>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.</p> <p>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).</p>