Bergman kernels and the pseudoeffectivity of relative canonical bundles

The main result of the present article is a (practically optimal) criterium for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. Our theorem has several applications in algebraic geometry; to start with, we obtain the natural analytic generalization of some semipositivity results due to E. Viehweg and F. Campana. As a byproduct, we give a simple and direct proof of a recent result due to C. Hacon--J. McKernan, S. Takayama and H. Tsuji concerning the extension of twisted pluricanonical forms. More applications will be offered in the sequel of this article.