Equivariant Euler characteristics of \mathscrM_{g, n}

2018/03/22 by Diaconu, Adrian
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.08349

Abstract

Let \mathscrM_{g, n} be the moduli space of n-pointed stable genus g curves, and let \mathscrM_{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 \mathbbMV generated by a stable \mathbbS-module V, allowing to effectively compute \mathbbS_n-equivariant Euler characteristics of \mathscrM_{g, n} in terms of \mathbbS_n'-equivariant Euler characteristics of \mathscrM_{g' , n'} with 0≤ g' ≤ g, \textrmmax\0, 3 - 2g' \ ≤ n' ≤ 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 \mathbbMV effective. To illustrate how the asymptotic expansion is used, we give formulas expressing the generating series of the \mathbbS_n-equivariant Euler characteristics of \mathscrM_{g, n}, for g = 0, 1 and 2, in terms of the corresponding generating series associated with \mathscrM_{g, n}.

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