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Book synopsis

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).

Book details

Book category Books in English Mathematics & science Mathematics Geometry

• Full title: Lectures on Vanishing Theorems
• Subtitle: Papers of a DMV-seminar on algebraic geometry, Schloß Reisensburg, October 13-19, 1991
• Author: Helene Esnault, Eckart Viehweg
• Language: English
• Binding: Paperback
• Number of pages: 166
• EAN: 9783764328221
• ISBN: 3764328223
• ID: 06930121
• Publisher: Birkhauser Verlag AG
• Weight: 350 g
• Dimensions: 244 × 170 × 10 mm
• Date of publishing: 01. December 1992

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