K\"ahler geometry on Hurwitz spaces

TL;DR

This paper explores the K"ahler geometry of Hurwitz spaces, focusing on the curvature of a generalized Weil-Petersson metric for simple branched coverings of the Riemann sphere.

Contribution

It introduces a framework for analyzing the curvature of the generalized Weil-Petersson metric on Hurwitz spaces using deformation theory of holomorphic maps.

Findings

Curvature properties of the generalized Weil-Petersson metric are characterized.

Deformation theory provides a new perspective on the geometry of Hurwitz spaces.

Abstract

We study the K\"ahler geometry of the classical Hurwitz space $\mathcal{H}^{n,b}$ of simple branched coverings of the Riemann sphere $\mathbb{P}^1$ by compact hyperbolic Riemann surfaces. A generalized Weil-Petersson metric on the Hurwitz space was recently introduced. Deformations of simple branched coverings fit into the more general framework of Horikawa's deformation theory of holomorphic maps, which we equip with distinguished representatives in the presence of hermitian metrics. In the article we will investigate the curvature of the generalized Weil-Petersson K\"ahler metric on the Hurwitz space.