Vanishing of !-restriction of constructible sheaves

Question

If $\mathcal F$ is a constructible sheaf (say of $\mathbb C$-modules) on a (real) manifold concentrated in degree $0$ and $i\colon Z \hookrightarrow X$ is a submanifold, can I say anything about $H^j(i^!\mathcal F)$ for $j > \operatorname{codim}_X Z$? Specifically, if $\mathcal F$ is constructible on $\mathbb R^n$ and $i$ is the embedding $\mathbb R^{n-1} \hookrightarrow \mathbb R^n$, does $H^j(i^! \mathcal F)$ vanish for $j > 1$?

(The analoguous statement for coherent sheaves is: if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$.)

Answer 1

I think the vanishing you want holds. For simplicity consider the case where $i$ is the inclusion of $Z = \{ z \}$ a point. (One should be able to reduce to this case by taking a normal slice.)

The question is local so we can replace $X$ by a small neighbourhood $U$ of $z$. Let $j$ denote the inclusion of $U - \{ z \}$. Consider the distinguished triangle $i_!i^!\mathcal{F} \to \mathcal{F} \to j_*j^*\mathcal{F} \to$. Now let $S^{n-1}_\epsilon$ be a sphere of radius $\epsilon$ around $z$. By the constructibility assumption the stalk of $j_*j^*\mathcal{F}$ at $z$ is equal, for small enough $\epsilon$, to the cohomology of $S^{n-1}_\epsilon$ with values in the restriction of $\mathcal{F}$. By standard vanishing theorems this vanishes in degrees $\ge n$. Hence the cohomology of $i_!i^!\mathcal{F}$ vanishes in degrees $> n$ as required.