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

Abstract

Let ${\overline{\mathcal{M}}}_{g,n}$ be the moduli space of $n$-pointed stable genus $g$ curves, and let ${\mathcal{M}}_{g,n}$ 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 ${\overline{\mathcal{M}}}_{g,n}$ in terms of ${\mathbb{S}}_{n^{\prime }}$-equivariant Euler characteristics of ${\mathcal{M}}_{g^{\prime },n^{\prime }}$ with $0\le g^{\prime }\le g,$ $\text{max}\{0,3-2g^{\prime }\}\le n^{\prime }\le 2(g-g^{\prime })+n.$ 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 ${\overline{\mathcal{M}}}_{g,n},$ for $g=0,1$ and $2,$ in terms of the corresponding generating series associated with ${\mathcal{M}}_{g,n}.$

arXiv comment16 pages

References in corpus (1)

• The trace of Hecke operators on the space of classical holomorphic Siegel modular forms of genus two