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?



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


