**Georg Schumacher**

Philipps University Marburg, Germany

February 19, 2015

**Kahler structure on Hurwitz spaces**:
The classical Hurwitz spaces, that parameterize compact Riemann
surfaces equipped with covering maps to ${mathbb P}_1$ with simple branch points, are extensively studied in the literature. We apply deformation theory, and investigate a K"ahler structure structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the covering map. The generalized Weil-Petersson form turns out to be the curvature of a Quillen metric on a determinant line bundle.?? The determinant line bundle extends to a compactification of the (generalized) Hurwitz space as a line bundle, and the Quillen metric yields a singular hermitian metric on the compactification so that a power of the determinant line bundle provides an embedding of the Hurwitz space in a projective space. (Joint work with Reynir Axelsson and Indranil Biswas).