K\"ahler structure on Hurwitz spaces

TL;DR

This paper studies the K"ahler geometry of Hurwitz spaces, introducing a generalized Weil-Petersson form linked to Quillen metrics, and explores their properties and compactifications.

Contribution

It introduces a generalized Weil-Petersson K"ahler form on Hurwitz spaces and relates it to Quillen metrics and determinant line bundles, extending the theory to arbitrary base curves.

Findings

Defined a generalized Weil-Petersson form as curvature of a Quillen metric.

Extended the determinant line bundle to compactifications of Hurwitz spaces.

Provided an embedding of Hurwitz spaces into projective space via the Quillen metric.

Abstract

The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to ${\mathbb P}_1$ of fixed numerical type with simple branch points, are extensively studied in the literature. We apply deformation theory, and present a study of the K\"ahler 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 K\"ahler 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 K\"ahler 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 ${\mathbb P}_1$. 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.