The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$

TL;DR

This work determines the eleventh cohomology of the Deligne–Mumford compactified moduli spaces $\bar{\mathcal{M}}_{g,n}$, proving $H^{11}(\bar{\mathcal{M}}_{g,n})=0$ unless $g=1$ and $n\ge 11$, while also establishing that even degree cohomology up to $k\le 12$ is pure Hodge–Tate and that point counts over finite fields are polynomially controlled. The authors combine the Arbarello–Cornalba boundary induction with a weight spectral sequence analysis of the boundary strata, and they compute the genus-1 contribution explicitly via cusp forms, identifying $H^{11,0}(\bar{\mathcal{M}}_{1,n})$ with a Specht module $V_{n-10,1^{10}}$. They then extend the vanishing to all $g\ge 2$ using a two-map complex on the boundary, reducing to base cases in genus $2$, and provide detailed graph-complex arguments to handle all cases. As consequences, they obtain precise point-count approximations, prove nonvanishing of higher odd cohomology in many genera and markings, and establish the existence of numerous non-tautological Chow classes, enriching the known structure of the tautological ring. The results support Langlands-inspired predictions for the cohomology of moduli spaces and reveal new interactions between arithmetic, geometry, and algebraic cycles in moduli theory.

Abstract

We prove that the rational cohomology group $H^{11}(\bar{\mathcal{M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\bar{\mathcal{M}}_{g,n})$ is pure Hodge-Tate for all even $k \leq 12$ and deduce that $\# \bar{\mathcal{M}}_{g,n}(\mathbb{F}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}(\bar{\mathcal{M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology.

Figures (5)

• Figure 1: The graph $\Gamma$ on the left and $\Gamma'$ on the right.
• Figure 2: The four flavors of $D_B$.
• Figure 3: Gluing $p_1$ and $q_1$.
• Figure 4: The graphs in Cases 1, 2, and 3.
• Figure 5: The graphs $\Gamma_{g,11-k}$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Theorem 1.5
• Corollary 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof