On totally geodesic submanifolds in the Jacobian locus

Abstract

We study submanifolds of Agthat 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 Mg↪ Agdue 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] ∈ Mgin 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 Agobtained from cyclic covers of ℙ1. 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.