3
$\begingroup$

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 $i > n$.)

$\endgroup$
1

1 Answer 1

7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.