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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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- Publisher:Birkhauser Basel
- Publication Date:06/12/2012
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- ISBN:9783034886000
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Download Now
- Format:PDF
- Publisher:Birkhauser Basel
- Publication Date:06/12/2012
- Category:
- ISBN:9783034886000