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Let $X$ be a scheme of finite type over $Spec(\mathbb{C})$. Let $X_{an}$ denote the associated complex analytic space. After fixing an isomorphism $\overline{\mathbb{Q}}_l\cong \mathbb{C}$, by $\S$6.1.2. of BBD (Asterisque 100) we know that there is fully faithfull functor

$$\mathcal{F}: D_c^b(X, \overline{\mathbb{Q}}_l) \rightarrow D_c^b(X_{an}, \mathbb{C}),$$

where the left hand side is the usual "derived category" of $l$-adic sheaves and the the right hand side is the usual derived category of complex sheaves with constructible (for an algebraic stratification) cohomology. As is well known this functor is not essentially surjective. However, on both sides we have two natural subcategories

$$Perv_{l}(X) \subset D_c^b(X, \overline{\mathbb{Q}}_l)$$

and

$$Perv_{\mathbb{C}}(X) \subset D_c^b(X_{an}, \mathbb{C}) $$ of l-adic Perverse sheaves and complex Perverse sheaves respectively. My question is this:

Does $\mathcal{F}$ induce an equivalence of categories between $Perv_{l}(X)$ and $Perv_{\mathbb{C}}(X)$?

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1 Answer

No.

If $L$ is a local system on a smooth variety $X$ of dimension $d$ then $L[d]$ is perverse. As suggested by BBD, if we take $L$ to be a rank $n$ local system of $\overline{\mathbb{Q}}_l$ vector spaces on $X_{an}$ then this is in the essential image of $\mathcal{F}$ iff the corresponding representation of $\pi_1(X_{an})$ preserves a lattice (by which I mean a rank $n$ $\mathcal{O}$-submodule, where $\mathcal{O}$ is the ring of integers in some finite extension of $\mathbb{Q}_p$). The reason for this is that the algebraic fundamental group is compact and any compact subgroup of $GL_n(\overline{\mathbb{Q}}_l)$ is contained in $GL_n(\mathcal{O})$, for some $\mathcal{O}$ as above.

This condition need not always hold: for example, one can consider representations of $\pi_1(X_{an})$ so that the eigenvalues of some element are algebraic but not $l$-adic integers.

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Just to clarify: Is the basic point that you'd like your local system to come from a representation of the full pro-finite fundamental group, and not just the topological fundamental group? –  Keerthi Madapusi Pera Oct 18 '11 at 17:57
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Yes. If ou just take $X=\mathbb{G}_m$, then the toplogical fundamental group is $\mathbb{Z}$, and a rank 1 local system of $\overline{Q}_\ell$ on $X_{an}$ corresponds to a representation of $\mathbb{Z}$ on $\overline{Q}_\ell$, i.e., an element $a$ of $\overline{Q}_\ell$. The local system comes from an étale local system on $X$ if and only if the representation extends to $\widehat{\mathbb{Z}}$, that is, if and only if $a\in\overline{\mathbb{Z}}_\ell^\times$. –  Alex Oct 18 '11 at 20:46
    
Sorry for the typos. Not sure what happened to the last formula. Anyway, let me reformulate it in words : $a$ is an $\ell$-adic unit. –  Alex Oct 18 '11 at 20:47
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