Equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$

Abstract

Let $\overline{\mathscr{M}}_{g, n}$ be the moduli space of $n$-pointed stable genus $g$ curves, and let $\mathscr{M}_{g, n}$ be the moduli space of $n$-pointed smooth curves of genus $g.$ In this paper, we obtain an asymptotic expansion for the characteristic of the free modular operad $\mathbb{M}\mathcal{V}$ generated by a stable $\mathbb{S}$-module $\mathcal{V},$ allowing to effectively compute $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$ in terms of $\mathbb{S}_{n'}$-equivariant Euler characteristics of $\mathscr{M}_{g'\!, n'}$ with $0\le g' \le g,$ $\textrm{max}\{0, 3 - 2g' \} \le n' \le 2(g - g') + n.$ This answers a question posed by Getzler and Kapranov by making their integral representation of the characteristic of the modular operad $\mathbb{M}\mathcal{V}$ effective. To illustrate how the asymptotic expansion is used, we give formulas expressing the generating series of the $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n},$ for $g = 0, 1$ and $2,$ in terms of the corresponding generating series associated with $\mathscr{M}_{g, n}.$