The Kawamata–Viehweg vanishing theorem for schemes

Abstract: In 1953, Kodaira proved what is now called the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations for complex projective varieties – in particular, the Kawamata–Viehweg vanishing theorem and its relative version due to Kawamata–Matsuda–Matsuki – have become indispensable tools in algebraic geometry over fields of characteristic zero, in particular in birational geometry and the minimal model program. However, while the goal in the minimal model program is to study birational equivalences between projective varieties, recent progress in the minimal model program has shown that it would be very useful to have a version of the Kawamata–Viehweg vanishing theorem that holds for schemes that are not necessarily projective varieties. Recently, I proved the relative Kawamata–Viehweg vanishing theorem for schemes of equal characteristic zero. My results are optimal given known counterexamples to vanishing theorems in positive and mixed characteristic, and have many applications to both algebraic geometry and commutative algebra. In this talk, I will discuss my vanishing theorem and several of its applications.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

The seminar takes place on Tuesdays and Thursdays via Zoom. Zoom passwords are given via mailing list on Fridays. To join the mailing list go to the website.

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Times vary to accommodate speakers time zones but times will be announced in GMT time.