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

Abstract

We study collections of subrings of $H^{\ast }({\overline{\mathcal{M}}}_{g,n})$ 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 $n\le 3$.

v3: 33 pages. Minor changes. Final version to appear in Forum Math. Pi

References in corpus (7)

• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• The Chow rings of the moduli spaces of curves of genus 7, 8, and 9
• Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves
• Cohomology of moduli spaces via a result of Chenevier and Lannes
• The eleventh cohomology group of ${\bar{\mathcal{M}}}_{g,n}$
• Weight 11 compactly supported cohomology of moduli spaces of curves
• Surjectivity of cycle maps for singular varieties

Cited by in corpus (1)

• Rational cohomology of ${\mathcal{M}}_{4,1}$