The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to P 1 of fixed numerical type with simple branch points, are extensively studied in the literature. We apply deformation theory, and present a study of the Kähler structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the meromorphic map. We introduce a generalized Weil-Petersson Kähler form on the Hurwitz space. This form turns out to be the curvature of a Quillen metric on a determinant line bundle. Alternatively, the generalized Weil-Petersson Kähler form can be characterized as the curvature form of the hermitian metric on the Deligne pairing of the relative canonical line bundle and the pull back of the anti-canonical line bundle on P 1 . Replacing the projective line by an arbitrary but fixed curve Y , we arrive at a generalized Hurwitz space with similar properties. 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.2000 Mathematics Subject Classification. 32G15, 14H10, 53C55.

Abstract. In the study of Teichmüller spaces the second variation of the logarithm of the geodesic length function plays a central role. So far, it was accessible only in a rather indirect way. We treat the problem directly in the more general framework of the deformation theory of Kähler-Einstein manifolds. For the first variation we arrive at a surprisingly simple formula, which only depends on harmonic Kodaira-Spencer forms. We also compute the second variation in the general case and then apply the result to families of Riemann surfaces. Again we obtain a simple formula depending only on the harmonic Beltrami differentials. As a consequence a new proof for the plurisubharmonicity of the geodesic length function on Teichmüller space and its logarithm together with upper estimates follow. The results also apply to the previously not known cases of Teichmüller spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available. We use our methods from [A-S], where the result was announced.

Abstract. In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmüller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available. Mathematics Subject Classification (2000). 32G15; 53C55, 53D30.

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