On the dimension of totally geodesic submanifolds in the Prym loci

TL;DR

This paper establishes improved bounds on the dimension of totally geodesic submanifolds within Prym loci of moduli spaces of polarized abelian varieties, adapting techniques from Torelli map studies.

Contribution

It provides new, tighter bounds on the dimension of totally geodesic submanifolds in Prym loci, extending previous results with novel adaptations of existing methods.

Findings

Improved bounds on the dimension of totally geodesic submanifolds in Prym loci.

Extension of techniques from Torelli map to Prym map cases.

Enhanced understanding of the geometric structure of Prym loci.

Abstract

In this paper we give a bound on the dimension 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. This improves the already known bounds. The idea is to adapt the techniques introduced by the authors in collaboration with A. Ghigi and G. P. Pirola for the Torelli map to the case of the Prym maps of (ramified) double covers.