Canonical 2-forms on the moduli space of Riemann surfaces

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 $g \geq 3$\mathrm{.}ThismeansanontrivialseconddeRhamcohomologyclasson${\mathbb M}_g$ 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 ${\mathbb{M}}_g$.