Are U(1) vortex moduli spaces on compact Kaehler surfaces (four manifolds) compact? If the base manifold is projective, will the moduli space be projective?
Previous search: on Kaehler surfaces, the perturbed Seiberg-Witten moduli space, under certain conditions, is the vortex moduli space, so compactness holds for these moduli spaces. But in case of vortices, not all line bundles generate spinc structures. What will happen in these cases?