**Georg Schumacher**

University of Marburg, Germany

September 30, 2010

**Positivity of relative canonical bundles and applications**:
Given an effectively parameterized family of canonically polarized
manifolds the K\"ahler-Einstein metrics on the fibers 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 bundle associated to the relative canonical bundle on the total space can be extended.
As an application we obtain a short analytic proof of the
quasi-projectivity 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.