Positivity of relative canonical bundles and applications

Given an effectively parameterized family of canonically polarized manifolds the Kahler-Einstein metrices on the fibres induce a hermitian metric on the relative canonical bundle.  We use a global elliptic equation to show that this metric is strictly positive and give estimates. For degenerating families it turns out that the curvature form on the total space can be controlled. By fiber integration it is shown that the generalized Weil-Petersson form on the base possesses an extension as a positive current. In this situation, the determinant line bundles associated to the relative canonical bundle on the total space can be extended.  As an application we obtain a short analytic proof of the quasiprojectivity of the moduli space ${\mathcal M}_{\mathrm {can}}$ of canonically polarized varieties. Further applications about curvature on higher direct image sheaves and hyperbolicity of moduli spaces are mentioned.