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

TL;DR

This paper extends the understanding of tautological rings in the cohomology of moduli spaces of curves, confirming key predictions and establishing new generation results for certain cohomology groups.

Contribution

It introduces new classes of subrings of cohomology that are stable under tautological operations and confirms predictions about Galois representations, Hodge structures, and generation of cohomology by tautological classes.

Findings

H^4(ar{M}_{g,n}) is generated by tautological classes for all g,n.

Confirmed predictions for Galois representations and Hodge structures at specific degrees.

Proved that the pure weight cohomology of M_{7,n} is generated by algebraic cycles for n ≤ 3.

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 $n \leq 3$.