Symplectic Fibrations and the Abelian Vortex Equations

The n ^th symmetric product of a Riemann surface carries a natural family of K?hler forms, arising from its interpretation as a moduli space of abelian vortices. We give a new proof of a formula of Manton–Nasir 10] for the cohomology classes of these forms. Further, we show how these ideas generalise to families of Riemann surfaces. These results help to clarify a conjecture of D. Salamon 13] on the relationship between Seiberg–Witten theory on 3–manifolds fibred over the circle and symplectic Floer homology.