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The proof of the Lefschetz Hyperplane theorem over $\mathbb{C}$ is well-known, and relies on Hodge theory in an important way. Does there exist an analogue of this theorem in positive characteristic, for an appropriate cohomology theory (crystalline? rigid?)?

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    $\begingroup$ For the etale topology, the weak Lefschetz theorem is a standard result, which can be found in the usual texts, but the hard Lefschetz theorem was only proved by Deligne as a result of his proof of the Weil conjectures (see the Wikipedia). $\endgroup$
    – abz
    Oct 1, 2013 at 15:36

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