H´el`eneEsnault Eckart Viehweg Lectures on Vanishing Theorems 1992 H´el`eneEsnault, Eckart Viehweg Fachbereich 6, Mathematik Universit¨at-Gesamthochschule Essen D-45117 Essen, Germany
[email protected] [email protected] ISBN 3-7643-2822-3 (Basel) ISBN 0-8176-2822-3 (Boston) c 1992 Birkh¨auserVerlag Basel, P.O. Box 133, CH-4010 Basel We cordially thank Birkh¨auser-Verlag for their permission to make this book available on the web. The page layout might be slightly different from the printed version. Acknowledgement These notes grew out of the DMV-seminar on algebraic geometry (Schloß Reisensburg, October 13 - 19, 1991). We thank the DMV (German Mathe- matical Society) for giving us the opportunity to organize this seminar and to present the theory of vanishing theorems to a group of younger mathemati- cians. We thank all the participants for their interest, for their useful comments and for the nice atmosphere during the seminar. Table of Contents Introduction . 1 1 Kodaira's vanishing theorem, a general discussion . 4 x 2 Logarithmic de Rham complexes . 11 x 3 Integral parts of Ql-divisors and coverings . 18 x 4 Vanishing theorems, the formal set-up. 35 x 5 Vanishing theorems for invertible sheaves . 42 x 6 Differential forms and higher direct images . 54 x 7 Some applications of vanishing theorems . 64 x 8 Characteristic p methods: Lifting of schemes . 82 x 9 The Frobenius and its liftings . 93 x 10 The proof of Deligne and Illusie [12] . 105 x 11 Vanishing theorems in characteristic p. 128 x 12 Deformation theory for cohomology groups .