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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?

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Surely you mean unitary connections on complex vector bundles on a complex manifold, for otherwise there is no underlying holomorphic structure. Since gauge theory is not only studied on complex manifolds, then could I ask you to please edit your question and state clearly your assumptions? Thanks. –  José Figueroa-O'Farrill Apr 19 '11 at 8:16
Many thanks for editing the question! –  José Figueroa-O'Farrill Apr 19 '11 at 11:00
You have been right, I should have ask my question more carefully. Thanks for your comments. –  Sebastian Apr 19 '11 at 11:26

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