This book, an extended collection of lectures delivered at "Schloss Reisensburg" during the DMV-Seminar "Algebraic Geometry, 1991", aims at presenting Kodaira's vanishing theorem and several generalizations in a way which is as algebraic as possible. We develop the theory of logarithmic de Rham complexes, the use of the corresponding spectral sequence, lifting properties for manifolds and their Frobenius morphisms in characteristic p>0 and the proof of the degeneration of the Hodge to de Rham spectral sequence with algebraic methods, due to P. Deligne and L. Illusie. We apply those methods to obtain vanishing theorems. Several typical applications and the generic theorems of M. Green and R. Lazarsfeld complete the picture. This exposition is self-contained and accessible to anyone with background in modern algebraic geometry. The necessary formalisms from cohomology theory are recalled in an appendix. Contents: Introduction • Kodaira's vanishing theorem, a general discussion • Logarithmic de Rham complexes • Integral parts of Q-divisors and coverings • Vanishing theorems, the formal set-up • Vanishing theorems for invertible sheaves • Differential forms and higher direct images • Some applications of vanishing theorems • Characteristic p methods: Lifting of schemes • The Frobenius and its liftings • The proof of Deligne and Illusie • Vanishing theorems in characteristic p. • Deformation theory for cohomology groups • Generic vanishing theorems • Appendix: Hypercohomology and spectral sequences • References

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