Bergman kernels and the pseudoeffectivity of relative canonical bundles

The main result of this article is a (practically optimal) criterion 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 40, 41 and F. Campana 6. As a byproduct, we give a simple and direct proof of a recent result due to C. Hacon and J. McKernan 16 and S. Takayama 29, 30 concerning the extension of twisted pluricanonical forms. More applications will be offered in 4, the sequel to this article