A Hitchin-Kobayashi Correspondence for Coherent Systems on Riemann Surfaces

A coherent system (E, V) consists of a holomorphic bundle plusa linear subspace of its space of holomorphic sections. Motivatedby the usual notion in geometric invariant theory, a notionof slope stability can be defined for such objects. In the paperit is shown that stability in this sense is equivalent to theexistence of solutions to a certain set of gauge theoretic equations.One of the equations is essentially the vortex equation (thatis, the Hermitian–Einstein equation with an additionalzeroth order term), and the other is an orthonormality conditionon a frame for the subspace VH⁰(E).