Gian Pietro Pirola
University of Pavia, Italy
April 11, 2019
On the local geometry of the Torelli locus: The Torelli map $t:M_g \to A_g$ 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 $g\gg 0$, according to the Coleman-Oort conjecture, they should not be contained in $t(M_g)$.
University of Pavia, Italy
April 11, 2019
On the local geometry of the Torelli locus: The Torelli map $t:M_g \to A_g$ 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 $g\gg 0$, according to the Coleman-Oort conjecture, they should not be contained in $t(M_g)$.