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.