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Title: Lectures on Vanishing Theorems ([EBook]) / by Hélène Esnault, Eckart Viehweg. Dewey Class: 500 Author: Esnault, Hélène., 1953- Added Personal Name: Viehweg, Eckart. author. Publication: Basel : Birkhäuser, 1992. Other name(s): SpringerLink (Online service) Physical Details: VIII, 166 pages : online resource. Series: DMV Seminar ;; 20 ISBN: 9783034886000 System details note: Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users) Summary Note: 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).: Contents note: § 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. ------------------------------ *** Es sind keine Exemplare vorhanden *** -----------------------------------------------
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