A Hitchin - Kobayashi correspondence for coherent systems on Riemann surfaces

A coherent system (ℰ, V) consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Motivated by the usual notion in geometric invariant theory, a notion of slope stability can be defined for such objects. In the paper it is shown that stability in this sense is equivalent to the existence of solutions to a certain set of gauge theoretic equations. One of the equations is essentially the vortex equation (that is, the Hermitian - Einstein equation with an additional zeroth order term), and the other is an orthonormality condition on a frame for the subspace V ⊂ H⁰ (ℰ).