Bergman kernels and the pseudoeffectivity of relative canonical bundles

Abstract

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