# What is the second fundamental form of moduli space?

Away from the hyperelliptic locus, the moduli of curves immerses in the moduli of principally polarized abelian varieties. The ambient space has a riemannian metric, so one can ask about the second fundamental form, the first-order deviation of the submanifold from being totally geodesic. What is this second fundamental form? Is anything known about it?

I think one could translate this into the language of algebraic geometry by using the Serre-Tate formal coordinates, which exist at each point of $A_g$. With respect to these coordinates, $M_g$ is not linear; what is its quadratic approximation? One could interpret this as a version of the Schottky problem, which suggests that existing solutions to it might be applicable.

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The following papers might be useful:

$(1)$ E. Colombo- G. Pirola- A. Tortora

"Hodge-Gaussian maps"

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 125–146.

$(2)$ E. Colombo - P. Frediani

"Siegel metric and curvature of the moduli space of curves"

Trans. Amer. Math. Soc. 362 (2010), no. 3, 1231–1246.

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