Kähler Geometry on Hurwitz Spaces

Abstract

The classical Hurwitz space ${\mathcal{H}}^{n,b}$ is a fine moduli space for simple branched coverings of the Riemann sphere ${\mathbb{P}}^1$ by compact hyperbolic Riemann surfaces. In the article we study a generalized Weil-Petersson metric on the Hurwitz space, which was introduced in [R. Axelsson et al., Manuscr. Math. 147, No. 1–2, 63–79 (2015; Zbl 1319.32012)]. For this purpose, Horikawa's deformation theory of holomorphic maps is refined in the presence of hermitian metrics in order to single out distinguished representatives. Our main result is a curvature formula for a subbundle of the tangent bundle on the Hurwitz space obtained as a direct image. This covers the case of the curvature of the fibers of the natural map ${\mathcal{H}}^{n,b}\to {\mathcal{M}}_g$.