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Given an effectively parameterized family of canonically polarized manifoldsthe Kaehler-Einstein metrics on the fibers induce a hermitian metric on therelative canonical bundle. We use a global elliptic equation to show that thismetric is strictly positive. For degenerating families we obtain a singularhermitian metric. Main application is an analytic proof of thequasi-projectivity of the moduli space of canonically polarized manifolds.Further applications in arXiv:1002.4858v2.

Positivity of relative canonical bundles for families of canonically polarized manifolds