Canonical 2-forms on the moduli space of Riemann surfaces

TL;DR

This paper reviews various canonical 2-forms on the moduli space of Riemann surfaces, emphasizing their uniqueness and potential for advancing the understanding of the space's secondary geometric structures.

Contribution

It consolidates different constructions of canonical 2-forms on the moduli space, facilitating future research in secondary geometry.

Findings

Second homology of ${ m M}_g$ is rank 1 for $g \,\geq\, 3$

Multiple canonical 2-forms exist on the moduli space

These forms differ but are related to the space's geometry

Abstract

As was shown by Harer the second homology of ${\mathbb M}_g$, the moduli space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \geq 3$. This means a nontrivial second de Rham cohomology class on ${\mathbb M}_g$ is unique up to constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts, and differ from each other. In this article we review some of constructions in order to provide materials for future research on "secondary geometry" of the moduli space ${\mathbb M}_g$.