On totally geodesic submanifolds in the Jacobian locus

TL;DR

This paper investigates totally geodesic submanifolds within the Jacobian locus, providing bounds on their dimensions based on gonality and genus, and analyzing specific submanifolds from cyclic covers to distinguish Shimura varieties.

Contribution

It introduces new bounds on the dimension of totally geodesic submanifolds in the Jacobian locus and characterizes which cyclic cover submanifolds are totally geodesic.

Findings

Upper bounds for dimensions depend on gonality and genus.

Many cyclic cover submanifolds are not totally geodesic.

Not all cyclic covers form Shimura varieties.

Abstract

We study submanifolds of A_g that are totally geodesic for the locally symmetric metric and which are contained in the closure of the Jacobian locus but not in its boundary. In the first section we recall a formula for the second fundamental form of the period map due to Pirola, Tortora and the first author. We show that this result can be stated quite neatly using a line bundle over the product of the curve with itself. We give an upper bound for the dimension of a germ of a totally geodesic submanifold passing through [C] in M_g in terms of the gonality of C. This yields an upper bound for the dimension of a germ of a totally geodesic submanifold contained in the Jacobian locus, which only depends on the genus. We also study the submanifolds of A_g obtained from cyclic covers of the projective line. These have been studied by various authors. Moonen determined which of them are Shimura varieties using deep results in positive characteristic. Using our methods we show that many of the submanifolds which are not Shimura varieties are not even totally geodesic.