On some differential-geometric aspects of the Torelli map

TL;DR

This paper surveys recent advances in understanding the extrinsic geometry of the Jacobian locus within the moduli space of abelian varieties, focusing on the Torelli map's second fundamental form and totally geodesic subvarieties.

Contribution

It compiles and discusses recent results on the geometry of the Jacobian locus, emphasizing the role of the second fundamental form and Hodge loci in this context.

Findings

Second fundamental form described as a multiplication map

Relation established between totally geodesic subvarieties and Hodge loci

Survey of various results on the geometry of the Jacobian locus

Abstract

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.