On the curvature of vortex moduli spaces

We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric products of the surface), we prove that, for genus $g>1$ , the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kähler metrics on certain symmetric products. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.