Torelli loci, product cycles, and the homomorphism conjecture for ${\mathcal{A}}_g$

Abstract

The tautological $\mathbb{Q}$-subalgebra ${\mathsf{R}}^{\ast }({\mathcal{A}}_g)\subset {\mathsf{CH}}^{\ast }({\mathcal{A}}_g)$ of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical $\mathbb{Q}$-linear projection operator $\mathsf{taut}:{\mathsf{CH}}^{\ast }({\mathcal{A}}_g)\to {\mathsf{R}}^{\ast }({\mathcal{A}}_g).$ We present here new calculations of intersection products of the Torelli locus in ${\mathcal{A}}_g$ with the product loci ${\mathcal{A}}_r\times {\mathcal{A}}_{g-r}\to {\mathcal{A}}_g$ for $r\le 3$. The results suggest that $\mathsf{taut}$ is a $\mathbb{Q}$-algebra homomorphism, at least for special cycles. We discuss a conjectural framework for this homomorphism property. Our calculations follow two independent approaches. The first is a direct study of the excess intersection geometry of the fiber product of the Torelli and product morphisms. The second recasts the geometry in terms of families Gromov-Witten classes, which are computed by a wall-crossing formula related to unramified maps. We define tautological projections of cycles on the fiber products ${\mathcal{X}}_g^s\to {\mathcal{A}}_g$ of the universal family. We compute these projections for a class of product cycles on ${\mathcal{X}}_g^s$ in terms of a determinant involving the universal theta divisors and Poincaré classes. Using Abel-Jacobi pullbacks of product cycles on ${\mathcal{X}}_g^s$ and their projections, we construct a new family of classes which we conjecture to lie in the Gorenstein kernels of the tautological rings ${\mathsf{R}}^{\ast }({\mathcal{M}}_{g,n}^{\mathrm{ct}})$. In particular, we construct nontrivial elements of the Gorenstein kernels of ${\mathsf{R}}^5({\mathcal{M}}_{5,2}^{\mathrm{ct}})$ and ${\mathsf{R}}^5({\mathcal{M}}_{4,4}^{\mathrm{ct}})$.

v2: 61 pages, added new Gorenstein kernel class and new Section 5.4