Lectures on Vanishing Theorems

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This work - an extended collection of lectures delivered at Schloss Reisenburg during the DMV-Seminar, Algebraic Geometry, 1991 - aims to present Kodaira’s vanishing theorem and several generalizations in a way which is as algebraic as possible. Together, the lectures 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. These methods are applied to obtain vanishing theorems. Several typical applications and the generic theorems of M. Green and R. Lazarsfeld complete the picture. This exposition should be accessible to anyone with a background in modern algebraic geometry. The necessary formalisms from cohomology theory are recalled in an appendix.