This question is about $\ell$-adic monodromy theorems for families over a number field. ($\ell$-adic analogues of Corollaries 6.2.8 and 6.2.9 in [BBD].)


Notation
========


$H$ denotes étale cohomology.


Let $f \colon X \to Y$ be a proper morphism of finite type schemes over $\mathbb{C}$.
Let $\mathcal{F}$ be in $\mathrm{D}^{\mathrm{b}}\_{\mathrm{c}}(X, \mathbb{Q}\_{\ell})$ (bounded, constructible), semi-simple of geometric origin.
Let $V \subset Y$ be a (Zariski) open over which $H^{i}f\_{\*}\mathcal{F}$ is locally constant. Take $y \in Y(\mathbb{C})$.


Question
========


The suggested global invariant cycle theorem:



> 
> **Q1.** If $V$ is connected, and $y \in V$, do we have a surjection
>  $$ H^{i}(X, \mathcal{F}) \twoheadrightarrow H^{i}(X\_{y}, \mathcal{F})^{\pi\_{1}(V, y)}$$
> 
> 
> 


And then the part that I am least sure about. (I am not even sure the formulation makes sense.) If I am not mistaken, one should replace open balls by henselian traits. Here is my try.


The suggested local invariant cycle theorem:



> 
> **Q2.** Let $B$ be the hensel localisation of $Y$ at $y$, and let $z$ be the generic point of $B$. Do we have a surjection
>  $$ H^{i}(X\_{y}, \mathcal{F}) \twoheadrightarrow H^{i}(X\_{z}, \mathcal{F})^{\pi\_{1}(B, z)} $$
> 
> 
> 


Remarks
=======


* I have the feeling that something like this should be true. Given the
theorems in [BBD], I think this should follow from their §6.1
“Principes”. Yet, I don't see how. Maybe this is because I don't
understand the proof of the decomposition theorem, nor how [6.2.8 and
6.2.9, BBD] are easy consequences of it.
* I am even more confident, because right after the decomposition
theorem [6.2.5, BBD] there is a remark that there is an étale
analogue.
* For personal applications, it is **Q2** that I am most interested in.




---


[BBD] — Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982). “Faisceaux pervers”. *Astérisque* (in French) (Société Mathématique de France, Paris) 100.



$\ell$-adic monodromy theorems (over $\mathbb{C}$)
The sages (for whom I am but an unworthy mouthpiece) say:


Tell the OP to look at 6.2.9 in Deligne’s Weil II paper, it’s quite close to answering 
what he is asking in his Q2.


