Vortices and the Abel-Jacobi map

In this paper we appeal to classical properties of Symd(Σ) to obtain new results for the moduli space metric. Our main tool is the Abel-Jacobi map, which maps Symd(Σ) into the Jacobian of Σ. Fibres of the Abel-Jacobi map are complex projective spaces, and the first theorem we prove states that near the Bradlow limit the moduli space metric restricted to these fibres is a multiple of the Fubini-Study metric. Additional significance is given to the fibres of the Abel-Jacobi map by our second result: we show that if Σ is a hyperelliptic surface, there exist two special fibres which are geodesic submanifolds of the moduli space. Even more is true: the Abel-Jacobi map has a number of fibres which contain complex projective subspaces that are geodesic.