The complex hyperbolic form as a Weil-Petersson form

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 Kähler form of the complex hyperbolic structure on the moduli space, as a kind of Weil-Petersson form.

arXiv commentv2, 29 pages. Final, various minor improvements according to referee's comment, to appear in Commun. Contemp. Math., the numbering of sections may be different from the published paper