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We generalize Tango’s theorem on the Frobenius map of the first cohomology groups to higher-dimensional algebraic varieties in characteristic p>0. As an application we construct counterexamples to Kodaira vanishing in higher dimension and prove the Ramanujam-type vanishing on surfaces which are not of general type when p≥5.
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Shigeru Mukai (2013) studied this question.
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