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 K\"ahler form of the complex hyperbolic structure on the moduli space, as a kind of Weil-Petersson form.Comment: 29 pages, all comments are welcome