Oberwolfach Seminars

Lectures on Vanishing Theorems

Authors: Esnault, Vieweg

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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).

Table of contents (14 chapters)

Table of contents (14 chapters)

Buy this book

eBook $69.99
price for USA in USD
  • ISBN 978-3-0348-8600-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $89.99
price for USA in USD
  • ISBN 978-3-7643-2822-1
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Lectures on Vanishing Theorems
Authors
Series Title
Oberwolfach Seminars
Series Volume
20
Copyright
1992
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0348-8600-0
DOI
10.1007/978-3-0348-8600-0
Softcover ISBN
978-3-7643-2822-1
Series ISSN
1661-237X
Edition Number
1
Number of Pages
VIII, 166
Topics