On a Riemann surface irreducible unitary connections on complex vector bundles of rank $n\geq 2$ are uniquley determined by their underlying holomorphic structure $\frac{1}{2}(\nabla+i*\nabla)$, where $*$ is the adjoint of the complex structure $J$ of the Riemann surface. This holomorphic structure is stable.

Are there any special examples, where this correspondence is made more explicit?