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Publication# Vanishing theorems in positive characteristic

Abstract

The topic of this thesis is vanishing theorems in positive characteristic. In particular, we use "the covering trick of Ekedahl" to investigate the vanishing of $H^1(X, \mathcal{O}_X(-D))$ for a big and nef Weil divisor $D$ on a normal projective variety with $-K_X$ nef. In dimension two, we show that on a surface of log del Pezzo type over a perfect field of characteristic $p>5$ this vanishing holds. More generally, using techniques of the \emph{Minimal model program} we prove the Kawamata--Viehweg vanishing theorem in this setting. We also construct a counter-example in characteristic five, showing that our result is optimal. We discuss the relationship (due to Hacon--Witaszek) between this vanishing theorem and properties of threefold klt-singularities. We investigate if a similar relationship exists between threefold lc-singularities and a certain vanishing theorem for higher direct images of elliptic fibrations. This leads to a counter-example to a theorem of Koll'ar over the complex numbers, in every positive characteristic.

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