What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwise pure?

Question

I'll freely admit that I have a rather hard time keeping straight different notions of purity of etale sheaves, and I think part of the problem is the lack of counterexamples.

For example, it's a theorem that the stalks of intersection cohomology sheaves (with coefficients in $\overline{\mathbb{Q}}_\ell$, say) are pure. But here, pure means "looks like the cohomology of a possibly singular projective variety" i.e., the weights are bounded above, but not below.

This is perhaps not so surprising: if, say, one takes constant coefficients, then the IC is a summand of the pushforward of a resolution of singularities, and so the stalks are summands of the cohomology of projective varieties.

But, if you do geometric representation theory, a funny thing happens. In practice, one seems to always get "pointwise purity" i.e. the stalk has the weights that one expects from the cohomology of a smooth projective variety. There are various geometric reasons for this (for example, if your resolution of singularity is symplectic, or if the fibers of your map have a paving/$\alpha$-partition by affines or other pure varieties), but it makes it hard for me to imagine anything else. I think my understanding of algebraic geometry would be improved by knowing an example of an intersection cohomology sheaf on (say) an affine variety which isn't pointwise pure.

What is a good example of such an intersection cohomology sheaf?

Answer 1

Here is an example. Sorry it's so complicated. (There's probably a simpler one, but my mind works in complicated ways, it seems.)

Consider a Siegel modular threefold $U$, i.e., a Shimura variety for the general symplectic group $GSp(4)$ with some level $n\geq 3$. (So that $U$ is smooth and quasi-projective over $\mathbb{Q}$.) Let $X$ be the Baily-Borel-Satake (aka minimal) compactification of $U$. Then $X$ has a stratification $X=U\cup X_1\cup X_2$, where $X_1$ is a finite disjoint union of modular curves and $X_2$ is a finite disjoint union of points (spectra of abelian extensions of $\mathbb{Q}$). Let's denote the inclusion of $U$ in $X$ by $j$, and define the intersection complex on $X$ by $IC=(j_{!*}\mathbb{Q}_\ell[2])[-2]$ (I like my intersection complexes to be concentrated in nonnegative degree). Then I say that the cohomology of the stalk of $IC$ at a point of $X_2$ is not pure (or pointwise pure, as you put it).

Before I do the messy calculation, let me give a heuristic reason. If we allowed ourselves to restrict everything to $Spec(\mathbb{C})$ and use $\mathbb{C}$-coefficients (which we can't because we want to look at weights of Frobenius), then we could use the calculation of the stalks of the intersection cohomology complex by Goresky, Harder, MacPherson and Nair in their article Local intersection cohomology of Baily-Borel compactifications (available on Goresky's webpage). In section 7 of this article, they work out the case of Siegel modular threefolds in some detail. In particular, they show (formula 7.7.2) that, if $x\in X_2$, then $IC_x$ has as a direct factor $H^*(X_\ell,\mathbb{C})$, where $X_\ell$ is a locally symmetric space associated to the linear part of one of the maximal Levi subgroups of $GSp(4)$; the linear part is in this case $GL(2)$. Morally, the cohomology of such "linear" locally symmetric spaces is made up of Artin motives, so the factor $H^*(X_\ell,\mathbb{C})$ should be of weight $0$ in every degree. But $H^1(X_\ell,\mathbb{C})$ contributes to $H^1(IC_x)$, so this vector space is not pure of weight $1$.

Okay, now for the formal proof. Unfortunately, there is no result as precise as the calculation of Goresky-Harder-MacPherson-Nair for the $\ell$-adic intersection complex on the variety $X$ over $\mathbb{Q}$, but, if we reduce modulo a prime of good reduction, then we can use the results of Morel's article Complexes pondérés sur les compactifications de Baily-Borel : le cas des variétés modulaires de Siegel. So now I will only consider the reductions of the various varieties modulo $p$, where $p$ is a big enough prime (here, it just means that $p$ does not divide the level).

I will need to introduce some notation. To make my life easier, I define the symplectic group with a symplectic form such that the intersection of $GSp(4)$ with the group of upper triangular matrices in $GL(4)$ is a Borel subgroup of $GSp(4)$ (say, an antidiagonal form). Let $N$ be the unipotent radical of this Borel subgroup. Let $N_2$ be the center of $N$, so that $N_2$ is the intersection of $GSp(4)$ and of the group $\left(\begin{matrix}1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix}\right)$

Let me also consider the following two cocharacters of $GSp(4)$ : $w_1:\lambda\longmapsto \left(\begin{array}{cccc}\lambda^2 & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & 1\end{array}\right)$ and $w_2:\lambda\longmapsto \left(\begin{array}{cccc}\lambda & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right)$

Note that the images of $w_1$ and $w_2$ normalize $N$ and $N_2$, so we get two actions of $\mathbb{G}_m$ on $H^*(Lie(N),\mathbb{Q}_\ell)$ and $H^*(Lie(N_2),\mathbb{Q}_\ell)$.

If $a$ and $b$ are integers, I will write $H^*(Lie(N),\mathbb{Q}_\ell)_{\geq a,<b}$ for the part of $H^*(Lie(N), \mathbb{Q}_\ell)$ where $w_2(\mathbb{G}_m)$ acts by characters $\lambda\longmapsto \lambda^k$ with $k\geq a$ and $w_1(\mathbb{G}_m)$ acts by characters $\lambda\longmapsto\lambda^k$ with $k<b$.

Likewise, I will write $H^*(Lie(N_2),\mathbb{Q}_\ell)_{\geq a}$ for the part of $H^*(Lie(N_2), \mathbb{Q}_\ell)$ where $w_2(\mathbb{G}_m)$ acts by characters $\lambda\longmapsto \lambda^k$ with $k\geq a$. (I don't care about the other action.)

Let as before $x$ be a point of $X_2$. From theorem 4.2.1 of Morel's article, I get a distinguished triangle : $H^*(IC_x)\longrightarrow K_1\longrightarrow K_2,$ where $K_1$ is finite direct sum of complexes of the form $H^*(\Gamma_\ell,H^*(Lie(N_2),\mathbb{Q}_\ell)_{\geq -3})$ and $K_2$ is a finite direct sum of complexes $H^*(Lie(N),\mathbb{Q}_\ell)_{\geq -3,<-1}$. Here, $\Gamma_\ell$ is an arithmetic subgroup of $GL(2,\mathbb{Q})$, where we see $GL(2)$ as a subgroup of $GSp(4)$ via the map $g\longmapsto\left(\begin{array}{cc}g & 0 \\0 & {}^tg^{-1}\end{array}\right)$; then $GL(2)$ acts by conjugation on $N_2$, hence it acts on $H^*(Lie(N_2),\mathbb{Q}_\ell)$, and it commutes with $w_2(\mathbb{G}_m)$, so it stabilizes $H^*(Lie(N_2),\mathbb{Q}_\ell)_{\geq -3}$. Note that the arithmetic subgroup $\Gamma_\ell$ could vary in the finite direct sum, but it won't matter.

(Note also that the theorem I cited only gives you an equality in a Grothendieck group. However, if instead of Morel's 3.3.5 (that it relies on), you instead invoke the stronger proposition 3.3.4 (ii), then you can see that this equality comes from a distinguished triangle.)

Another thing we need to know is that the (Frobenius) weights on $K_1$ and $K_2$ are given by minus the weights of $w_2(\mathbb{G}_m)$; that is, if $w_2(\mathbb{G}_m)$ acts on some part of $K_1$ or $K_2$ by the character $\lambda\longmapsto\lambda^k$, then the (Frobenius) weight of this part is $-k$. (This is proved for example in an article of Pink, and the reference is given in proposition 2.1.4 of the article of Morel I'm citing.)

Now I need to look a little more closely at the action of my two tori on the first cohomology groups of $Lie(N)$ and $Lie(N_2)$. They both act trivially on the $H^0$ groups, so in particular $H^0(K_1)$ has (Frobenius) weight $0$ and $H^0(K_2)=0$ (because $H^0(Lie(N_2),\mathbb {Q}_\ell)_{\geq -3}=H^0(Lie(N_2),\mathbb {Q}_\ell)$ and $H^0(Lie(N),\mathbb {Q}_\ell)_{\geq -3,<-1}=0$).

The torus $w_2(\mathbb{G}_m)$ acts by $\lambda\longmapsto\lambda$ on $Lie(N_2)$, so it acts by $\lambda\longmapsto \lambda^{-1}$ on $H^1(Lie(N_2),\mathbb{Q}_\ell)$, and $H^1(Lie(N_2),\mathbb{Q}_\ell)_{\geq -3}=H^1(Lie(N_2),\mathbb{Q}_\ell)$.

Let $V=N/N_2$; this is a $1$-dimensional unipotent group. Using the spectral sequence $E_2^{pq}=H^p(Lie(V),H^q(Lie(N_2),\mathbb{Q}_\ell))\Longrightarrow H^{p+q}(Lie(N),\mathbb{Q}_\ell)$ and the fact that $H^p(Lie(V),\ )$ vanishes for $p>1$, we get an exact sequence $0\longrightarrow H^1(Lie(V),H^0(Lie(N_2),\mathbb{Q}_\ell))\longrightarrow H^1(Lie(N),\mathbb{Q}_\ell)\longrightarrow H^0(Lie(V),H^1(Lie(N_2),\mathbb{Q} _\ell))$. On the third term, $w_1(\mathbb{G}_m)$ acts by $\lambda\longmapsto\lambda^k$ for $0\leq k\leq 2$ and $w_2(\mathbb{G}_m)$ acts by $\lambda\longmapsto \lambda^{-1}$. On the first term, $w_1(\mathbb{G}_m)$ acts by $\lambda \longmapsto\lambda^{-1}$ and $w_2(\mathbb{G}_m)$ acts trivially. (This follows by looking at the weights of the tori in $Lie(N_2)$ and $Lie(V)$.) In particular, we find that $H^1(Lie(N),\mathbb{Q}_\ell)_{\geq -3,<-1}$ injects into $H^0(Lie(V),H^1(Lie(N_2),\mathbb{Q}_\ell))$, and this implies that $H^1(K_2)$ has (Frobenius) weight $1$.

The distinguished triangle above gives us an exact sequence $0=H^0(K_2)\longrightarrow H^1(IC_x)\longrightarrow H^1(K_1)\longrightarrow H^1(K_2)$. Remembering the formula for $K_1$ above, we see that $H^1(K_1)$ is a finite direct sum of groups of the form $H^0(\Gamma_\ell,H^1(Lie(N_2),\mathbb{Q}_\ell))$ and $H^1(\Gamma_\ell,H^0(Lie(N_2),\mathbb{Q}_\ell))$ (with $\Gamma_\ell$ as above). The first summand has (Frobenius) weight $1$, and the second summand has (Frobenius) weight $0$.

But we have seen that $H^1(K_2)$ has weight $1$. So the second summand is sent to $0$ in $H^1(K_2)$, and $H^1(IC_x)$ contains a finite direct sum of groups of the form $H^1(\Gamma_\ell,H^0(Lie(N_2),\mathbb{Q}_\ell))$, in particular in contains a part of weight $0$, so it is not pure of weight $1$. (Note that all we did was prove an algebraic version of a very small part of the result of Goresky-Harder-MacPherson-Nair.)

Answer 2

This isn't really a bona fide answer. It's more a series of thoughts, which perhaps you or someone else can complete. Take a normal surface singularity $(X,x)$, whose resolution (which exists in any characteristic) consists of a cycle of smooth curves (with at least one having nonzero genus). The point is that the dual graph should have some first homology. I suspect that this may be an example of what you're looking for. My impression is that the singularities that show up in representation theory tend to be rational and so quite far from this.

Since I'm a complex geometer, let me say things over $\mathbb{C}$, but I suspect it works $\ell$-adically. (I'm trying to learn the $\ell$-adic stuff, but I haven't quite got there.) Set $IC= j_{!*}\mathbb{Q}[2]$, where $j:X-x\to X$ is the inclusion. The stalk of $H^{-1}(IC_x)$ is the first cohomology of the link of the singularity, and this won't be pure as a Hodge structure. The assumptions ensure that both $W_0$ and $W_1$ are nonzero.

Answer 3

Here is an example along the lines of Donu's suggestion:

We work over a (large) finite field $k$. Let $C \subset \mathbb{P}^2$ be an irreducible plane curve of degree $d$ and geometric genus $g > 0$ having at least one node. For example, the curve given by the equation $$y^2 = (x^2-1)(x-a_1)\cdots(x-a_{d-2})$$ with $d>4$, all $a_i$ distinct and nonzero and $char(k) \neq 2$.

Blow up $d^2 + 1$ distinct smooth points on $C$ to get a smooth surface $X$ and let $D$ be the strict transform of $C$. We have $D\cdot D = -1$ since $C\cdot C = d^2$. By Artin's contractability criterion, we can contract $D$ to obtain a normal surface $Y$ with a singular point $p$. Now $H^1(D, \mathbb{Q}_{\ell})$ has weights $0$ and $1$, the $0$ coming from the node and the $1$ since the geometric genus is positive. By proper base change and the decomposition theorem, it follows that $IC_Y$ is not pointwise pure (at $p$).