In Deligne's paper on his first proof of the Weil conjectures, we have the following result.

Theorem 5.10 (Kazhdan-Margulis). L'image de $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ est ouverte.

This theorem says the monodromy group of a Lefschetz pencil of odd fiber dimension is "as big as possible".

My question is, where has this proof been written down in the literature? I'm finding the discussion in Deligne to be quite terse, and I searched the papers of Kazhdan and Marguilis and I couldn't find a paper which establishes this result.

  • 4
    $\begingroup$ my commenting on this will be a case of "fools rush in where angels fear to tread"; I will try anyway. The monodromy representation from the topological $\pi _1(U)$ actually goes into $Sp(E/E\cap E^{\perp},\mathbb{Z})$; if you grant that the image is Zariski dense in the symplectic group, then by going to $l$-adic completion, you get that the image of $\pi (U)$ has open closure in $Sp(E/E\cap E^{\perp},\mathbb{Z}_l))$ (by a result of Nori and Weisfeiler); in other words, the map of the algebraic fundamental group $\pi _1(U)^*$ into the $l$-adic symplectic group has open image. $\endgroup$ Oct 20 '16 at 6:36

The proof of this result is worked out in detail in page 250 (theorem 7.5) of the book

  • Eberhard Freitag, Reinhardt Kiehl "Etale Cohomology and the Weil Conjecture"

You can preview that page in particular here.


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