On the local geometry of the Torelli locus

Abstract: 

The Torelli map t:M_g->A_g  gives an immersion (outside the hyperelliptic locus) of the moduli space of complex curves of genus g  into  the moduli space of principally polarized abelian varieties of dimension g. We study the local geometry of this immersion by means of the natural riemannian (orbifold) structure induced on A_g from Siegel space. In particular two methods to give a bound on the dimension of the totally geodesic subvarieties of A_g contained in M_g are discussed. The first one (Colombo-Frediani-Ghigi) uses the second fundamental form associated to the Torelli immersion and the second one (Ghigi-P-Torelli) uses instead a sort of local Fujita decomposition along geodesics. We recall that the Shimura varieties are (algebraic) totally geodesic subvarieties of A_g and  for g>>0, according to the Coleman-Oort conjecture, they should not be contained in t(M_g) .