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

## TL;DR

This paper provides a geometric interpretation of the second fundamental form of the period map of curves, leading to improved bounds on the dimensions of totally geodesic subvarieties within the Torelli locus and hyperelliptic Torelli locus.

## Contribution

It introduces a geometric interpretation of the second fundamental form and uses it to refine upper bounds on the dimensions of certain totally geodesic subvarieties in A_g.

## Key findings

- Dim Y < 2g if g is even
- Dim Y < 2g+1 if g is odd
- Dim Z < g+2 for hyperelliptic Torelli locus

## 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 A_g generically contained in the Torelli locus obtained in [3], [7]. We get dim Y < 2g if g is even, dim Y < 2g+1 if g is odd. We also study totally geodesic subvarieties Z of A_g generically contained in the hyperelliptic Torelli locus and we show that dim Z < g+2.