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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?

It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety. Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for …