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The weight two compactly supported Euler characteristic of moduli spaces of curves

Sam Payne, Thomas Willwacher

TL;DR

This paper provides an explicit generating function for the weight two, S_n-equivariant compactly supported Euler characteristics of moduli spaces of curves, extending the known weight-zero results by deploying graph complexes tied to operadic methods. Central to the approach is the identification of gr_2 H_c^•(\mathcal M_{g,n}) with the cohomology of the graph complex X_{g,n} (Payne–Willwacher), and the computation of its equivariant Euler characteristics via decorated graphs, Kontsevich-type graph operads, and a careful plethystic calculus of symmetric sequences. The authors derive a detailed formula for the all-genus generating function \omega_2 in the ring of formal symmetric power series, including logarithmic and Laurent-monomial contributions in inhomogeneous power-sum variables, with corrections for unstable low-genus cases. The results combine deep operadic machinery with explicit combinatorial and asymptotic analyses (e.g., polygamma expansions) to yield both a theoretical framework and practical computability for small genus and $n$. The work advances understanding of weight-2 information in the mixed Hodge structure of moduli spaces and provides a roadmap for interpreting terms combinatorially through graph generators, with potential implications for related embedding-Calculus and tropical-geometry perspectives.

Abstract

We derive a formula for the generating function for the weight two compactly supported $\mathbb S_n$-equivariant Euler characteristics of the moduli spaces of curves $\mathcal M_{g,n}$, using graph complexes and calculations inspired by operadic methods.

The weight two compactly supported Euler characteristic of moduli spaces of curves

TL;DR

This paper provides an explicit generating function for the weight two, S_n-equivariant compactly supported Euler characteristics of moduli spaces of curves, extending the known weight-zero results by deploying graph complexes tied to operadic methods. Central to the approach is the identification of gr_2 H_c^•(\mathcal M_{g,n}) with the cohomology of the graph complex X_{g,n} (Payne–Willwacher), and the computation of its equivariant Euler characteristics via decorated graphs, Kontsevich-type graph operads, and a careful plethystic calculus of symmetric sequences. The authors derive a detailed formula for the all-genus generating function \omega_2 in the ring of formal symmetric power series, including logarithmic and Laurent-monomial contributions in inhomogeneous power-sum variables, with corrections for unstable low-genus cases. The results combine deep operadic machinery with explicit combinatorial and asymptotic analyses (e.g., polygamma expansions) to yield both a theoretical framework and practical computability for small genus and . The work advances understanding of weight-2 information in the mixed Hodge structure of moduli spaces and provides a roadmap for interpreting terms combinatorially through graph generators, with potential implications for related embedding-Calculus and tropical-geometry perspectives.

Abstract

We derive a formula for the generating function for the weight two compactly supported -equivariant Euler characteristics of the moduli spaces of curves , using graph complexes and calculations inspired by operadic methods.
Paper Structure (23 sections, 13 theorems, 117 equations, 2 figures)

This paper contains 23 sections, 13 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main result
  3. Relations to the topological and weight zero Euler characteristics
  4. Numerical computations
  5. Preliminaries
  6. Notation and conventions
  7. Symmetric sequences and products
  8. Equivariant Euler characteristics
  9. Examples of symmetric sequences and their equivariant Euler characteristics
  10. Graph complexes associated to moduli spaces of curves
  11. The graph complexes $G^{(g,n)}$ and weight zero compactly supported cohomology of ${\mathcal{M}}_{g,n}$
  12. Two variants of $G^{(g,n)}$
  13. Graph complexes with one special vertex
  14. Proof of Proposition \ref{['prop:iHom Gr']}
  15. The graph complexes $X_{g,n}$ and weight two compactly supported cohomology of ${\mathcal{M}}_{g,n}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

The generating function for $\mathbb{S}$-equivariant weight $2$ compactly supported Euler characteristics of moduli spaces of curves is

Figures (2)

  • Figure 1: The equivariant Euler characteristic of $\mathop{\mathrm{gr}}\nolimits_2H_c^\bullet({\mathcal{M}}_{g,n})$ for $g\leq 10$ and $n\leq 5$, computed using Mathematica and Sage from Theorem \ref{['thm:eulerchar']} and expressed in terms of Schur functions.
  • Figure 2: Plots of $\log(1+|\chi_2({\mathcal{M}}_{g})|)$ and the sign of $\chi_2({\mathcal{M}}_{g})$ for $g \leq 200$, on the left and right, respectively.

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Conjecture 1.3
  • Theorem 3.1: CGP2
  • Proposition 3.2
  • Theorem 3.3: Kontsevich KMotives, Lambrechts-Volic LambrechtsVolic
  • Lemma 3.4
  • proof
  • proof : Proof of Proposition \ref{['prop:iHom Gr']}
  • Theorem 3.5: Theorem 1.1 of PayneWillwacher
  • ...and 11 more