On some differential-geometric aspects of the Torelli map

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.

To appear on Boll. UMI, special volume in memory of Paolo de Bartolomeis