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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.

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  • $\begingroup$ For Q1 I guess you meant to assume that y is in V. I believe the answer is no in general. I guess you know that it's true if V=Y so you're in the situation of the usual invariant cycle theorem. $\endgroup$ – Dan Petersen Sep 5 '14 at 13:43
  • $\begingroup$ @DanPetersen — Thanks for your comment. You're absolutely right: $y \in V$. Fixed. I need the case where $f$ is not smooth. I want to understand the monodromy at singular fibres, so $V \ne Y$. $\endgroup$ – jmc Sep 5 '14 at 13:53
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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.

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    $\begingroup$ If it lies within your possibilities, please pass my honour and respect to the sages, for in times of great need, their infinite wisdom was of great help to this mere mortal. $\endgroup$ – jmc Sep 6 '14 at 11:16
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    $\begingroup$ On a more mathematical note: I didn't know that the last pages of Weil II dealt with arbitrary algebraically closed fields. Awesome! Moreover, 6.2.12 is “quite close to answering what he is asking in his” Q1. For that reason, I think this is a good time to hit the “Accept” button. $\endgroup$ – jmc Sep 6 '14 at 11:19
  • $\begingroup$ @jmc I will pass your thanks to the sages, who are, indeed, surpassing wise. $\endgroup$ – Igor Rivin Sep 6 '14 at 23:08

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