On Shimura Subvarieties of the Torelli Locus and Ramified Prym Maps

The purpose of the thesis is two-fold: to investigate the existence of totally geodesic subvarieties of the moduli space of principally polarized abelian varieties, Ag, contained in the Jacobian locus and to study the geometry of certain positive dimensional fibres of some ramified Prym maps. Totally geodesic subvarieties constitute a useful tool to study the extrinsic geometry of the Jacobian locus inside Ag and they are involved in the rather famous Coleman-Oort conjecture. Furthermore, they motivate our interest in Prym maps. Indeed it turns out that certain positive dimensional fibres represent a good place to look for totally geodesic subvarieties.