Tautological and non-tautological cycles on the moduli space of abelian varieties

Abstract

The tautological Chow ring of the moduli space ${\mathcal{A}}_g$ of principally polarized abelian varieties of dimension $g$ was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from ${\mathcal{A}}_g$ to the moduli space ${\mathcal{M}}_g^{\mathrm{ct}}$ of genus $g$ of curves of compact type, we prove that the product class $[{\mathcal{A}}_1\times {\mathcal{A}}_5]\in {\mathsf{CH}}^5({\mathcal{A}}_6)$ is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the the tautological ring ${\mathsf{R}}^{\ast }({\mathcal{M}}_6^{\mathrm{ct}})$ in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring ${\mathsf{R}}^{\ast }({\mathcal{M}}_6^{\mathrm{ct}})$ has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[{\mathcal{A}}_1\times {\mathcal{A}}_5]$. More generally, the Torelli pullback of the difference between $[{\mathcal{A}}_1\times {\mathcal{A}}_{g-1}]$ and its tautological projection always lies in the Gorenstein kernel of ${\mathsf{R}}^{\ast }({\mathcal{M}}_g^{\mathrm{ct}})$. The product map ${\mathcal{A}}_1\times {\mathcal{A}}_{g-1}\to {\mathcal{A}}_g$ is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all $g$ are presented.

v3: 57 pages, to appear in Invent. Math