Zoom link:  https://princeton.zoom.us/j/91248028438

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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 have become indispensable tools in algebraic geometry over fields of characteristic zero, in particular in birational geometry and the minimal model program. Even in this context, it is often necessary to work with schemes of finite type over power series rings, and a fundamental problem has been the lack of Kodaira-type vanishing theorems. We prove Kodaira's vanishing theorem and some generalizations in the more general context of excellent schemes of equal characteristic zero, resolving conjectures of Boutot and Kawakita.