Lectures on Vanishing Theorems

ISBN-13: 9783764328221 / Angielski / Miękka / 1992 / 166 str.

Helene Esnault;Esnault;Vieweg

Opis

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).

Spis treści

§ 1 Kodaira’s vanishing theorem, a general discussion.- § 2 Logarithmic de Rham complexes.- § 3 Integral parts of Q-divisors and coverings.- § 4 Vanishing theorems, the formal set-up.- § 5 Vanishing theorems for invertible sheaves.- § 6 Differential forms and higher direct images.- § 7 Some applications of vanishing theorems.- § 8 Characteristic p methods: Lifting of schemes.- § 9 The Frobenius and its liftings.- § 10 The proof of Deligne and Illusie [12].- § 11 Vanishing theorems in characteristic p.- § 12 Deformation theory for cohomology groups.- § 13 Generic vanishing theorems [26], [14].- Appendix: Hypercohomology and spectral sequences.- References.