Compact Subvarieties of the Moduli Space of Complex Abelian Varieties

TL;DR

The paper resolves the maximal dimensions of compact subvarieties in the moduli space ${ m A}_g$ of principally polarized abelian varieties and determines the maximal compact-dimension through a general point, linking these to Hodge-generic and special subvarieties via Ax–Schanuel. By combining Shimura variety theory, decoupled and non-decoupled symplectic representations, and Satake classifications, the authors derive an exact formula for ${ m mdc}({ m A}_g)$ and prove ${ m mdc}_{vg}({ m A}_g)=g-1$, with a nuanced description of the maximal subvarieties arising from decomposable loci. The work further yields precise results for moduli of curves of compact type, including bounds for ${ m mdc}({ m M}_g^{ m ct})$, and illuminates the indecomposable and Jacobian loci by relating their geometry to the decoupled-special subvarieties that dominate the maximal-dimension cases. The findings have broad implications for understanding the geometric and arithmetic structure of moduli spaces, their special subvarieties, and the Torelli map on compact-type curves.

Abstract

We determine the maximal dimension of compact subvarieties of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties of dimension $g$, and the maximal dimension of a compact subvariety through a very general point of $\mathcal{A}_g$. This also allows us to draw some conclusions for compact subvarieties of the moduli space of complex curves of compact type.

Theorems & Definitions (96)

• Example 1.1
• Remark 1.2
• Remark 1.3
• Theorem A
• Theorem B
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Corollary C
• Definition 2.1