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. §0 Introduction In this article our primary goal is to establish some positivity results concerning the twisted relative canonical bundle of projective morphisms. LetX and Y be non-singular projective manifolds, and let p : X → Y be a surjective projective map, whose relative dimension is equal to n. Consider also a line bundle L over X , endowed with a metric h = e, such that the curvature current is semipositive (unless explicitly stated otherwise, all the metrics in this article are allowed to be singular). We denote by I(h) the multiplier ideal sheaf of h (see e.g. [10], [19]). Let Xy be the fiber of p over the point y ∈ Y , which is assumed to be general enough such that the restriction of the metric h to Xy is well defined (i.e. it is not identically +∞), and such that y is not a critical value of p. A point which satisfies these two requirements will be called ”generic” throughout the present article. Under these circumstances, the space of (n, 0) forms L-valued on Xy which belong to the multiplier ideal sheaf of the restriction of the metric h is endowed with a natural L–metric as follows ‖u‖y := ∫