Totally geodesic subvarieties in the moduli space of curves

Abstract

In this paper, we study totally geodesic subvarieties [Formula: see text] of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for [Formula: see text]. We prove that if [Formula: see text] is generically contained in the Torelli locus, then [Formula: see text].

On the dimension of totally geodesic submanifolds in the Prym loci

“…The purpose of this paper is to improve the estimates obtained in [4,5] on the maximal dimension of a germ of a totally geodesic submanifold of the moduli space of polarised abelian varieties of a given dimension, which is contained in the Prym locus of a (possibly) ramified double cover. The idea is to adapt to the Prym case the technique developed in [6] and [9] to give a bound on the maximal dimension of a germ of a totally geodesic submanifold contained in the Torelli locus (see also [11]).…”

On the Jacobian locus in the Prym locus and geodesics

“…compatible with the metric structure (see e.g. a classical reference [Gri71] or some more recent references [Ghi18], [GPT19]).…”

“…One proves separately that such a geodesic is not locally contained in J g (Jacobian case) and then even in P g+1 (Prym case). The argument in the Jacobian case is a straightforward application of some previous works: especially from [GPT19], but some preparatory results are contained in [PT19], [GST19] and [GAT18] (see Section 3, Lemma 3.1). The argument in the Prym case is much more complicated and new, involving degeneration techniques (see Section 4).…”

Some evidence for the Coleman–Oort conjecture

“…This expectation is also motivated by another stronger expectation originating from the point of view of differential geometry: special subvarieties are totally geodesic with respect to the locally symmetric (orbifold) metric on A g (the one coming from the Siegel space). If one believes that j(M g ) bears no strong relation to the ambient geometry of A g , in particular that it is very curved inside A g , then it is natural to expect that j(M g ) contains generically no totally geodesic subvarieties, and in particular no Shimura subvarieties (see [9,17,19] for results in this direction).…”