Holomorphic forms and non-tautological cycles on moduli spaces of curves

TL;DR

The paper proves the existence of non-tautological algebraic cohomology classes on moduli spaces $\mathcal{M}_{g,n}$ for infinitely many $(g,n)$ by examining double cover loci arising from admissible degree-2 maps to lower-genus curves. It leverages holomorphic forms on $\overline{\mathcal{M}}_{h,k}$ and pure weight cohomology to produce nontrivial, non-boundary-constrained contributions to diagonal components, thereby showing certain double cover cycles are non-tautological both in the interior and, via boundary analysis, on compactified moduli spaces. The main result yields non-tautological classes on $\mathcal{M}_{g,2m}$ for $g+m=12$ or $g+m\ge 16$, and extends to non-tautological classes on $\mathcal{M}^{rt}_{g,n}$ with odd marked points, broadening the previously known finite list of such classes and linking to prior work of Graber–Pandharipande, van Zelm, and Lian. This advances understanding of the tautological vs non-tautological structure of $H^*(\overline{\mathcal{M}}_{g,n})$ and $H^*(\mathcal{M}_{g,n})$ through a framework that combines admissible covers, diagonal pullbacks, holomorphic forms, and weight filtrations.

Abstract

We prove, for infinitely many values of $g$ and $n$, the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular, when $n=0$, our results show that there exist non-tautological algebraic cohomology classes on $\mathcal{M}_g$ for $g=12$ and all $g \geq 16$. These results generalize the work of Graber--Pandharipande and van Zelm, who proved that the classes of particular loci of bielliptic curves are non-tautological and thereby exhibited the only previously-known non-tautological class on any $\mathcal{M}_g$: the bielliptic cycle on $\mathcal{M}_{12}$. We extend their work by using the existence of holomorphic forms on certain moduli spaces $\overline{\mathcal{M}}_{g,n}$ to produce non-tautological classes with nontrivial restriction to the interior, via which we conclude that the classes of many new double-cover loci are non-tautological.

Figures (2)

• Figure 1: The pairs $(g,n)$ for which all algebraic classes on $\mathcal{M}_{g,n}$ are tautological (indicated by circles in the figure) all follow from previous work. Aside from the eleven pairs on the blue line, all cases where non-tautological algebraic classes on $\mathcal{M}_{g,n}$ exist (indicated by $\color{red}{\times}$'s in the figure) are new to the current work.
• Figure 2: An illustration of the commutative diagram \ref{['commutative diagram: gluing maps']}.

Theorems & Definitions (27)

• Theorem A
• Remark
• Remark
• Theorem B
• Remark
• Definition 1.1
• Remark 1.2
• Proposition 1.3
• Lemma 1.4
• proof