2 POSITIVITY OF RELATIVE CANONICAL BUNDLES AND APPLICATIONS

Abstract

Abstract. Given a family f: X → S of canonically polarized manifolds, the unique Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle KX/S. We use a global elliptic equation to show that this metric is strictly positive on X, unless the family is infinitesimally trivial. For degenerating families we show that the curvature form on the total space can be extended as a (semi-)positive closed current. By fiber integration it follows that the generalized Weil-Peters-son form on the base possesses an extension as a positive current. We prove an extension theorem for hermitian line bundles, whose curvature forms have this property. This theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space. As an application the quasi-projectivity of the moduli spaceMcan of canonically polarized varieties follows. The direct images Rn−pf∗Ω p X/S(K⊗mX/S), m> 0, carry natural hermitian metrics. We prove an explicit formula for the curva-ture tensor of these direct images. We apply it to the morphisms SpTS → Rpf∗ΛpTX/S that are induced by the Kodaira-Spencer map and obtain a differential geometric proof for hyperbolicity properties of Mcan.