Computing Verdier duality (perverse sheaves)

Question

I have a question concerning the method of computing Verdier duality.

Let $X$ be a smooth complex algebraic variety and $\Bbb C_X$ be the constant sheaf for $\Bbb C$. Let $\omega^{\bullet}_X=a^!_X \Bbb C\in D^b(\Bbb C_X)$ where $a_X:X\rightarrow \text{pt}$ is the unique morphism from $X$ to the one-point space pt. It is the dualizing complex of $X$ and $\omega_X=\Bbb C_X[2\text{dim}X]$. (because we assumed that $X$ is a complex manifold) The Verdier dual $\Bbb D_X(F^{\bullet})$ is defined by $\Bbb D_X(F^{\bullet})=R\mathcal{Hom}_{\Bbb C_X}(F^{\bullet},\omega^{\bullet}_X)\in D^b(\Bbb C_X)$.

1. Although it seems to be pretty clear, I don't know how to compute the Verdier dual because I don't know how to construct the injective resolution of $\omega^{\bullet}_X=\Bbb C_X[2\text{dim}X]$. Which injective resolution do we typically use when computing $\Bbb D_X$? In addition, is there some sufficient conditions for a module to be injective in $\text{Mod}(\Bbb C_X)$?

2. Also to check if $F^{\bullet}$ a perverse sheaf, we should check the dimension of the support of $H^i(\Bbb D_X(F^{\bullet}))$, and therefore we should know the stalk $(H^i(\Bbb D_X(F^{\bullet})))_x$ for each $x\in X$. Since $(\Bbb C_X)_x=\Bbb C$, my naive guess is that $(H^i(\Bbb D_X(F^{\bullet})))_x=H^i(RHom_{\Bbb C} (F^{\bullet}_x,\Bbb C[2\text{dim} X]))$ where $F^{\bullet}_x$ is the complex of $\Bbb C$-vector spaces consisting of stalks of each objects in $F^{\bullet}$ at $x$. Is it right?

Thank you.

Answer 1

In my opinion, one should view the injective resolution definition of $\mathbb{D}$ as fundamentally not something computable, or at least not something you should want to compute. In general, I would say I “know” a (complex) of sheaves if I know its stalks/costalks, and understand how it looks on strata. Resolving by injectives doesn’t really illustrate any of this, so using the definitions of the derived functors makes things harder, from my perspective.

Luckily there are plenty of tools one can use beyond the definition to get a handle on such things, most of which are formal. I found Intersection Cohomology II by Goresky, Macpherson and Deligne to be fantastic for spelling out the whole (topological) picture.

A specific useful result in there (middle of page 91) is that the stalks of the hyper cohomology at a point $x$ is the hyper cohomology of the complex restricted to a distinguished neighbourhood of the point, which dualises to a description for the costalks (${i_x}i^!$ for $i_x$ point inclusion) as the compactly supported cohomology on this neighbourhood.

For instance, this gives you the stalks of the dualising sheaf itself, its stalks measure the cohomology of compactifications of the links around all points (which explains why it’s so simple for smooth spaces, but complicated for singular ones).