The complex hyperbolic form as a Weil-Petersson form

TL;DR

This paper establishes a new equality between two symplectic forms on the moduli space of punctured spheres, interpreting the complex hyperbolic form as a Weil-Petersson form through Euclidean metrics.

Contribution

It introduces a novel interpretation of the complex hyperbolic form as a Weil-Petersson form on the moduli space of punctured spheres.

Findings

New equality between symplectic forms

Interpretation of complex hyperbolic form as Weil-Petersson form

Application to moduli space of punctured spheres

Abstract

For the moduli space of the punctured spheres, we find a new equality between two symplectic forms defined on it. Namely, by treating the elements of this moduli space as the singular Euclidean metrics on a sphere, we give an interpretation of the complex hyperbolic form, i.e. the K\"ahler form of the complex hyperbolic structure on the moduli space, as a kind of Weil-Petersson form.