Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$

TL;DR

This work develops a framework of semi-tautological extensions (STEs) and the Chow–Künneth generation Property (CKgP) to study motivic structures in $H^*(\bar{\mathcal{M}}_{g,n})$. It introduces inductive criteria to extend tautological generation via boundary maps and base-case control, proving predictions of Chenevier–Lannes for degrees $k\le 15$, including the tautological generation of $H^4$ and the pure weight descriptions of $H^{13}$ and $H^{15}$ across genera. A central innovation is proving CKgP for genus $7$ with up to three marked points using a combination of hyperelliptic/trigonal, tetragonal, and pentagonal loci, notably via Mukai’s pentagonal construction and the orthogonal Grassmannian. The results yield precise decompositions of high-degree cohomology in terms of genus $1$ data and establish that certain even cohomology and low-degree homology groups are tautological, with significant implications for the structure of Galois representations and Hodge structures in the cohomology of moduli spaces. Overall, the paper provides a cohesive inductive strategy, grounded in geometric constructions, to determine motivic and tautological features of the cohomology of moduli of curves in low to intermediate degrees.

Abstract

We study collections of subrings of $H^*(\overline{\mathcal{M}}_{g,n})$ that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and contain the tautological subrings. Such extensions of tautological rings are well-suited for inductive arguments and flexible enough for a wide range of applications. In particular, we confirm predictions of Chenevier and Lannes for the $\ell$-adic Galois representations and Hodge structures that appear in $H^k(\overline{\mathcal{M}}_{g,n})$ for $k = 13$, $14$, and $15$. We also show that $H^4(\overline{\mathcal{M}}_{g,n})$ is generated by tautological classes for all $g$ and $n$, confirming a prediction of Arbarello and Cornalba from the 1990s. In order to establish the final bases cases needed for the inductive proofs of our main results, we use Mukai's construction of canonically embedded pentagonal curves of genus 7 as linear sections of an orthogonal Grassmannian and a decomposition of the diagonal to show that the pure weight cohomology of $\mathcal{M}_{7,n}$ is generated by algebraic cycle classes, for $n \leq 3$.

Figures (3)

• Figure 1: The argument in the proof of Theorem \ref{['thm:fgSTE']} shows that $W_{17}H^{17}(\mathcal{M}_{g,n})/(\Phi^{17}_{g,n} + RH^{17}(\mathcal{M}_{g,n}))$ vanishes for $(g,n)$ outside the gray shaded region. Note that this quotient does not vanish for $(g,n)$ equal to $(1,17)$ and $(2,14)$, which are pictured by purple dots.
• Figure 2: Selected Künneth components of pullbacks of the $891$ classes to boundary divisors. In the chart, each row represents several rows of a block matrix. The number of rows in each group is listed to its left. The two half-edges glued to make a boundary divisor (pictured across the top) are labeled $q$ on the left vertex and $p$ on the right vertex.
• Figure 3: The graph $\Gamma$ on the left and $\Gamma'$ on the right.

Theorems & Definitions (103)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Conjecture 1.8
• Theorem 1.9
• Theorem 1.10