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

Frediani, Paola; Pirola, Gian

doi:10.1090/proc/15291

On the geometry of the second fundamental form of the Torelli map
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by Paola Frediani and Gian Pietro Pirola PDF

Proc. Amer. Math. Soc. 149 (2021), 1011-1024 Request permission

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 $Y$ of $\mathsf {A}_g$ 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 $\dim Y \leq 2g-1$ if $g$ is even, $\dim Y \leq 2g$ if $g$ is odd. We also study totally geodesic subvarieties $Z$ of $\mathsf {A}_g$ generically contained in the hyperelliptic Torelli locus and we show that $\dim Z \leq g+1$.

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