The Kodaira vanishing theorem and generalizations

by Poon, Wai-hoi

(Uncorrected OCR) Abstract of thesis entitled THE KODAIRA VANISHING THEOREM AND GENERALIZATIONS submitted by Bobby Wai-hoi POON for the Degree of Master of Philosophy at the University of Hong Kong in June 2002 This thesis is a survey on the cohomology vanishing theorem by Kodaira saying that for a compact Kahler manifold M with canonical line bundle Km and positive holomorphic line bundle L on M, the cohomology group Hq(M, Km?L) = 0 for all q > 0. The basic terminology in the theory of complex manifolds is introduced and the proof of the Kodaira Vanishing Theorem is followed by the use of Bochner formulae and identification of the cohomology group to the space of harmonic forms. Two generalizations of the Kodaira Vanishing Theorem are studied: Nadel? Vanishing Theorem on Multiplier Ideal Sheaves and Kawamata-Viehweg? Vanishing Theorem on nef and big Q-divisors. By making use of Hormander? L2-estimates for the ?-equation, Nadel established his Vanishing Theorem. Given a complex projective manifold M and a holomorphic line bundle L equipped with a singular Hermitian metric h satisfying the positivity condition that the curvature form of L is no less than a positive multiple of the Kahler metric on M, Nadel? Vanishing Theorem asserts that Hq(M,Km ? L ? I(h)) = 0 for all q > 0, where I(h) is the multiplier ideal sheaf defined by h. On the other hand, Kawamata-Viehweg? Vanishing Theorem is more algebraic in nature. The essential basic notions in algebraic geometry are developed for the proof of the theorem. Kawamata-Viehweg? Vanishing Theorem states that for any complex projective manifold M, L is a holomorphic line bundle which is nef and big, then Hq(M, Km ? L) = 0 for all q > 0. In this thesis the analytic and algebraic techniques for proving Kodaira? Vanishing Theorem and its generalization are studied in details and contrasted with one another. Some applications of Kodaira? Vanishing Theorem are also given.