Lectures on Vanishing Theorems

1. Front Matter

   Pages i-vii

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2. Introduction

   • Hélène Esnault, Eckart Viehweg

   Pages 1-3

3. Kodaira’s vanishing theorem, a general discussion

   • Hélène Esnault, Eckart Viehweg

   Pages 4-10

4. Logarithmic de Rham complexes

   • Hélène Esnault, Eckart Viehweg

   Pages 11-18

5. Integral parts of Q-divisors and coverings

   • Hélène Esnault, Eckart Viehweg

   Pages 18-35

6. Vanishing theorems, the formal set-up

   • Hélène Esnault, Eckart Viehweg

   Pages 35-42

7. Vanishing theorems for invertible sheaves

   • Hélène Esnault, Eckart Viehweg

   Pages 42-54

8. Differential forms and higher direct images

   • Hélène Esnault, Eckart Viehweg

   Pages 54-64

9. Some applications of vanishing theorems

   • Hélène Esnault, Eckart Viehweg

   Pages 64-82

10. Characteristic p methods: Lifting of schemes

    • Hélène Esnault, Eckart Viehweg

    Pages 82-93

11. The Frobenius and its liftings

    • Hélène Esnault, Eckart Viehweg

    Pages 93-105

12. The proof of Deligne and Illusie [12]

    • Hélène Esnault, Eckart Viehweg

    Pages 105-128

13. Vanishing theorems in characteristic p

    • Hélène Esnault, Eckart Viehweg

    Pages 128-132

14. Deformation theory for cohomology groups

    • Hélène Esnault, Eckart Viehweg

    Pages 132-136

15. Generic vanishing theorems [26], [14]

    • Hélène Esnault, Eckart Viehweg

    Pages 137-146

16. Back Matter

    Pages 147-166

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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).

• Divisor
• algebra
• algebraic geometry
• cohomology
• deformation theory
• manifold

• Fachbereich 6, Mathematik, Universität GH Essen, Essen, Germany

  Hélène Esnault, Eckart Viehweg

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