Annulation de la cohomologie pour les fibres semi-positifs

From two general theorems (1 and 2) we deduce vanishing theorems in situations where only finiteness is usually obtained. We give in particular simple proofs of theorems of H. Grauert and O. Riemenschneider [6], and of K. Takegoshi [10]. Following an idea of [11], we also obtain results in the relative case; as a consequence, we get a generalization of Nakano's theorem [9].