Lectures on Vanishing Theorems

✍ Scribed by Hélène Esnault, Eckart Viehweg (auth.)

• Publisher: Birkhäuser Basel
• Year: 1992
• Tongue: English
• Leaves: 172
• Series: DMV Seminar 20
• Edition: 1
• Category: Library

✦ 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).

✦ Table of Contents

Front Matter....Pages i-vii
Introduction....Pages 1-3
Kodaira’s vanishing theorem, a general discussion....Pages 4-10
Logarithmic de Rham complexes....Pages 11-18
Integral parts of Q-divisors and coverings....Pages 18-35
Vanishing theorems, the formal set-up....Pages 35-42
Vanishing theorems for invertible sheaves....Pages 42-54
Differential forms and higher direct images....Pages 54-64
Some applications of vanishing theorems....Pages 64-82
Characteristic p methods: Lifting of schemes....Pages 82-93
The Frobenius and its liftings....Pages 93-105
The proof of Deligne and Illusie [12]....Pages 105-128
Vanishing theorems in characteristic p ....Pages 128-132
Deformation theory for cohomology groups....Pages 132-136
Generic vanishing theorems [26], [14]....Pages 137-146
Back Matter....Pages 147-166

✦ Subjects

Science, general