A coherent system ([Escr ], 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([Escr ]).