On Kollár's vanishing theorems. I (Q752107)

Let \(\pi\) : \(X\to Y\) be a smooth morphism of projective nonsingular varieties. One gives a vanishing theorem involving certain complexes of sheaves on Y which appear in the construction by Deligne and Zucker of Hodge structures on the Leray factors \(H^ s(Y,R^ t\pi_*{\mathbb{C}})\). In fact the theorem concerns an arbitrary variation of Hodge structures over Y. The results extend Kollár's (and Kodaira's) vanishing theorem.