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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 K X /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-Petersson 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 space M can of canonically polarized varieties follows.The direct images R n−p f * Ω p X /S (K ⊗m X /S ), m > 0, carry natural hermitian metrics. We prove an explicit formula for the curvature tensor of these direct images. We apply it to the morphisms S p T S → R p f * Λ p T X /S that are induced by the Kodaira-Spencer map and obtain a differential geometric proof for hyperbolicity properties of M can .

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We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kähler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.2000 Mathematics Subject Classification. 53C07, 32L05.

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