# Example of an intersection complex not concentrated in a single degree

I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful.

I want to construct an example of an intersection complex not concentrated in a single (natural) cohomology degree. Reading BBD, it seems the definition of intermediate extension needs to be made in the derived category, even to discuss intermediate extension of constant sheaves. So I think I can find an example of an open inclusion $$j: U_0 \hookrightarrow X_0$$ such that $$j_{!*} \bar{\mathbb{Q}}_{\ell} [d]$$ is not concentrated in degree $$d$$. I'm looking for a simplest example, but I'm having trouble verifying my work so far. So I would also appreciate if someone could point out any glaring errors in my reasoning (and lack thereof).

The first few attempts I made all seem to have $$R^1 j_! \bar{\mathbb{Q}}_{\ell} = 0$$, and so $$Rj_! \bar{\mathbb{Q}}_{\ell} [d] = {}^p j_!\bar{\mathbb{Q}}_{\ell}[d]$$; so $${}^pj_! \bar{\mathbb{Q}}_{\ell} [d] \hookrightarrow {}^p j_* \bar{\mathbb{Q}}_{\ell} [d]$$; hence $$j_{!*} (\bar{\mathbb{Q}}_{\ell} [d]) = j_!(\bar{\mathbb{Q}}_{\ell}) [d]$$.

In particular, the above seems that to hold whenever $$X_0$$ is smooth and $$j: U_0 \hookrightarrow X_0$$ is the inclusion of dense open. So this is not the right direction.

Looking now at singular varieties, the first two examples that come to mind are $$C_0 = \mathrm{Proj} (\mathbb{F}_q[S,T,U]/(T^2U-S^3))$$ (projective cubic curve with a cusp) and $$C'_0 = \mathrm{Proj} (\mathbb{F}_q[S,T,U]/(T^2U - S^3 - S^2U))$$ (projective cubic curve with a node). Note the nonsingular loci $$C_{ns, 0} \cong \mathbb{A}^1_0$$ and $$C'_{ns, 0} \cong \mathbb{G}_{m, 0}$$. (Assume $$\mathrm{char}(\mathbb{F}_q) > 2$$ for $$C'_0$$.)

But in the case of $$C_0$$, taking $$j: C_{ns,0} \hookrightarrow C_0$$ to be the inclusion of the nonsingular locus, it appears to me that $$Rj_!$$ is exact. In particular, the stalk at a geometric point $${\bar{x}}$$ lying over the node $$x \in C_0$$ $$(R^1 j_! \bar{\mathbb{Q}}_{\ell})_{\bar{x}} = \lim_{\to} H^1 (U, j_! \bar{\mathbb{Q}}_{\ell}) \overset{(a)}{=} \lim_{\to} H^1_c (U \times_{C_{0}} C_{ns, 0}, \bar{\mathbb{Q}}_{\ell}) \overset{(b)}{\cong} H^1_c (\mathbb{A}^1, \bar{\mathbb{Q}}_{\ell}),$$ where the limit is taken over étale $$U \to C_0$$ over $$\bar{x}$$. Then we have $$H^1_c(\mathbb{A}^1, \bar{\mathbb{Q}}_{\ell})$$ vanishes by Poincaré dualtiy as $$H^1 (\mathbb{A}^1, \bar{\mathbb{Q}}_{\ell}) = 0$$. (I think $$(a)$$ holds by definition of $$H^*_c$$, and $$(b)$$ I can't justify.) So, assuming every link in this chain holds, we have $$j_! = j_{!*}$$, and I have not found my example.

But I believe—if my reasoning is at all accurate for $$C_0$$—that I have found an example in $$j': C'_{ns, 0} \hookrightarrow C'_0$$. Repeating the argument above, with $$x' \in C'_0$$ the self-intersection point, $$(R^1 j'_! \bar{\mathbb{Q}}_{\ell})_{\bar{x}'} = \lim_{\to} H^1 (U, j'_! \bar{\mathbb{Q}}_{\ell}) = \lim_{\to} H^1_c (U \times_{C'_0} C'_{ns, 0}, \bar{\mathbb{Q}}_{\ell}) \cong H^1_c (\mathbb{G}_{m}, \bar{\mathbb{Q}}_{\ell}).$$ In this case, we have $$H^1 (\mathbb{G}_{m}, \bar{\mathbb{Q}}_{\ell}) = \bar{\mathbb{Q}}_{\ell}(-1)$$ (this is my understanding after reading Milne's and de Jong's notes on étale cohomology), and so $$(R^1 j'_! \bar{\mathbb{Q}}_{\ell})_{\bar{x}'} = \bar{\mathbb{Q}}_{\ell}(1) \ne 0$$. Since we have determined now that $$j'_!$$ is not exact, we need to calculate $${}^p j'_!$$, $${}^p j'_*$$, and finally calculate $$j'_{!*}$$. Should I keep going? Am I on the right track? Have I made glaring errors? Is there a reason $$(b)$$ should hold? What can I read to speed up my progress on these questions? I've read BBD and Kiehl-Weissauer, and a couple of less formal notes on perverse sheaves, and I've seen precious few examples in any detail. I recognize I haven't read the entire literature, so does anyone know where I should look next?

• Tiny comment that isn't really that helpful: the singularity on $C_0$ is a cusp, not a node. Jul 22 '20 at 23:41
• The most beautiful exposition of perverse sheaves is (arxiv.org/abs/0712.0349). There are examples there of intersection cohomology complexes supported in multiple degrees. Jul 23 '20 at 13:37

Sorry I haven't read your entire question, which is a bit long. This is really just an extended comment to address the "where I should look next?" part. Suppose $$X$$ has an isolated singularity $$x$$, and $$j:U\to X$$ is the smooth complement. Then the formula on top of page 60 of BBD would simplify to $$j_{!*}\overline{\mathbb{Q}}_\ell[n]= (\tau_{\le n-1}\mathbb{R} j_* \overline{\mathbb{Q}}_\ell)[n]$$ where $$n=\dim X$$ and I'm using middle perversity. Now let $$X$$ be a sufficiently complicated singularity, a cone over an elliptic curve will do. Then this won't be a translate of a sheaf. Look at the stalk at $$x$$, it will have cohomology in 2 degrees.
• Should the index of the truncation be $n-1$ rather than $n$? Jul 23 '20 at 3:23