1
$\begingroup$

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of sheaves) $D(V)$. My question is: how to find an $e$ as large as possible so that $C\in D^{t\ge e}$? The problem is that for a singular $V$ and an embedding $i:T\to V$ it seems difficult to compute $i^!C$ (as well as the Verdier dual of $C$).

In particular, I would like to obtain an interesting estimate for $e$ in the case when $V$ is smooth in 'large' codimension (say, that is not much smaller than the dimension of $V$), but whose singularities are 'very bad'.

$\endgroup$

1 Answer 1

2
$\begingroup$

Let $n$ be the dimension of $V$. I assume that by "constant sheaf", you meant "constant sheaf shifted in degree $-n$", so that if $V$ is smooth, $C$ is indeed perverse.

Let us call $e$ the best possible integer as in your question. Then we generally have $e\geq -n$, and if the variety is smooth we have $e=0$. If I understand your question, you would like an estimate that tells you that, if the singular locus has "big" codimension, then e is close to $0$.

However, let me try to build an example of a $n$-dimensional variety with a $0$-dimensional singular locus for which $e=2-n$. If it works, then it rules out the possibility of any interesting estimate as you wish.

Start with a smooth $n-1$-dimensional subvariety $X$ in projective space $P^{N}$ and let $V$ be the associated cone in $A^{N+1}$. The singular locus is the origin $o$. Let $U$ be the open smooth complement of $o$ and let $i:o\hookrightarrow V$ and $j: U\hookrightarrow V$ be the corresponding immersions.

There is a distinguished triangle $$ i^! C \longrightarrow i^* C\longrightarrow i^* Rj_* C \longrightarrow $$ and an isomorphism $$ i^* Rj_* C \simeq R\Gamma(U,C) .$$ The latter comes from the $\mathbb G_m$ action on $V$ (see Lemma 4.5 in Kazhdan-Lusztig paper "Schubert varieties and Poincaré duality").

Taking $C=\mathbb Q_\ell[n]$ , we get $H^{2-n}(i^!C)\simeq H^1(U,\mathbb Q_\ell)$. So it remains to find $X$ such that $H^1(U)\neq 0$.

But $U$ is a $\mathbb G_m$-torsor over $X$ and thus is the complement of the zero section of some line bundle on $X$. Therefore we have a triangle $$ R\Gamma(X)[-2] \longrightarrow R\Gamma(X) \longrightarrow R\Gamma(U) \longrightarrow $$ where the first map is the Lefschetz operator on $X$ associated to $U$.

This yields an isomorphism $H^1(X)\simeq H^1(U)$, so it remains to take $X$ with $H^1(X)\neq 0$.

Hope I didn't write too much nonsense.

$\endgroup$

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.