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Lectures on Vanishing Theorems

  • Hélène Esnault
  • Eckart Viehweg

Part of the DMV Seminar book series (OWS, volume 20)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Hélène Esnault, Eckart Viehweg
    Pages 1-3
  3. Hélène Esnault, Eckart Viehweg
    Pages 4-10
  4. Hélène Esnault, Eckart Viehweg
    Pages 11-18
  5. Hélène Esnault, Eckart Viehweg
    Pages 18-35
  6. Hélène Esnault, Eckart Viehweg
    Pages 35-42
  7. Hélène Esnault, Eckart Viehweg
    Pages 42-54
  8. Hélène Esnault, Eckart Viehweg
    Pages 54-64
  9. Hélène Esnault, Eckart Viehweg
    Pages 64-82
  10. Hélène Esnault, Eckart Viehweg
    Pages 82-93
  11. Hélène Esnault, Eckart Viehweg
    Pages 93-105
  12. Hélène Esnault, Eckart Viehweg
    Pages 105-128
  13. Hélène Esnault, Eckart Viehweg
    Pages 128-132
  14. Hélène Esnault, Eckart Viehweg
    Pages 132-136
  15. Hélène Esnault, Eckart Viehweg
    Pages 137-146
  16. Back Matter
    Pages 147-166

About this book

Introduction

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).

Keywords

Divisor algebra algebraic geometry cohomology deformation theory manifold

Authors and affiliations

  • Hélène Esnault
    • 1
  • Eckart Viehweg
    • 1
  1. 1.Fachbereich 6, MathematikUniversität GH EssenEssenGermany

Bibliographic information