O ct 2 00 7 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 and S. Takayama 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 -possibly singularmetric h = e, such that the curvature current is semi-positive. We denote by I(h) the multiplier ideal sheaf of h (see e.g. [10], [21], [24]). Let Xy be the fiber of p over a point y ∈ Y , such that y is not a critical value of p. We also assume at first that the restriction of the metric h to Xy is not identically +∞. 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