# Totally geodesic subvarieties in the moduli space of curves

## TL;DR

This paper investigates the dimensions of totally geodesic subvarieties within the moduli space of abelian varieties, establishing bounds when these subvarieties are contained in the Torelli locus for genus g ≥ 4.

## Contribution

It provides a new upper bound on the dimension of such subvarieties, advancing understanding of the geometry of the moduli space and Torelli locus.

## Key findings

- Dimension bound:  Y ext{ is in Torelli locus, then } ext{dim } Y \u2264 (7g - 2)/3.
- Applicable for genus g  4.
- Enhances knowledge of the structure of totally geodesic subvarieties.

## Abstract

In this paper we study totally geodesic subvarieties $Y \subset \mathsf{A}_g$ of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for $g\geq 4$. We prove that if $Y$ is generically contained in the Torelli locus, then $\dim Y \leq (7g -2)/3$.