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Introduction M. Kodaira's vanishing theorem, saying that the inverse of an ample invert ible sheaf on a projective complex manifold X has no cohomology below the dimension of X and its generalization, due to Y. Akizuki and S. Nakano, have been proven originally by methods from differential geometry ([39J and [1]). Even if, due to J.P. Serre's GAGA-theorems [56J and base change for field extensions the algebraic analogue was obtained for projective manifolds over a field k of characteristic p = 0, for a long time no algebraic proof was known and no generalization to p > 0, except for certain lower dimensional manifolds. Worse, counterexamples due to M. Raynaud [52J showed that in characteristic p > 0 some additional assumptions were needed. This was the state of the art until P. Deligne and 1. Illusie [12J proved the degeneration of the Hodge to de Rham spectral sequence for projective manifolds X defined over a field k of characteristic p > 0 and liftable to the second Witt vectors W2(k). Standard degeneration arguments allow to deduce the degeneration of the Hodge to de Rham spectral sequence in characteristic zero, as well, a re sult which again could only be obtained by analytic and differential geometric methods beforehand. As a corollary of their methods M. Raynaud (loc. cit.) gave an easy proof of Kodaira vanishing in all characteristics, provided that X lifts to W2(k).
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Table of contents (14 chapters)
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Front Matter
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Back Matter
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Bibliographic information
Book title: Lectures on Vanishing Theorems
Authors: Hélène Esnault, Eckart Viehweg
Series title: Oberwolfach Seminars
DOI: https://doi.org/10.1007/978-3-0348-8600-0
Publisher: Birkhäuser Basel
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eBook packages: Springer Book Archive
Copyright information: Springer Basel AG 1992
Softcover ISBN: 978-3-7643-2822-1Published: 01 December 1992
eBook ISBN: 978-3-0348-8600-0Published: 06 December 2012
Series ISSN: 1661-237X
Series E-ISSN: 2296-5041
Edition number: 1
Number of pages: VIII, 166
Topics: Science, Humanities and Social Sciences, multidisciplinary