A universal construction for moduli spaces of decorated vector bundles over curves

Let X be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation ρ : GL(r)−→ GL(V). Then, one can associate to every vector bundle E of rank r over X a vector bundle Eρ with fibre V . We would like to study triples (E,L,φ) where E is a vector bundle of rank r over X , L is a line bundle over X , and φ : Eρ −→ L is a non-trivial homomorphism. This set-up comprises well-known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert-Mumford criterion, and establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.