On the geometry of the second fundamental form of the Torelli map

Abstract

In this paper we give a geometric interpretation of the second fundamental form of the period map of curves and we use it to improve the upper bounds on the dimension of a totally geodesic subvariety Y Y of A g \mathsf {A}_g generically contained in the Torelli locus obtained by Elisabetta Colombo, Paola Frediani, and Alessandro Ghigi [Internat. J. Math. 26 (2015), no. 1, 1550005] and A. Ghigi, P. Pirola, and S. Torelli (to appear on Communications in Contemporary Mathematics, https:// doi.org/10.1142/S0219199720500200). We get dim ⁡ Y ≤ 2 g − 1 \dim Y \leq 2g-1 if g g is even, dim ⁡ Y ≤ 2 g \dim Y \leq 2g if g g is odd. We also study totally geodesic subvarieties Z Z of A g \mathsf {A}_g generically contained in the hyperelliptic Torelli locus and we show that dim ⁡ Z ≤ g + 1 \dim Z \leq g+1 .