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As Charlie Frohman says, for curves this is the Narasimhan-Seshadri correspondonce. For Kahler manifolds of higher dimensions it is called the Hitchin-Kobayashi correspondence, proved by Donaldson-Uhlenbeck-Yau. You search for a Hermitian metric on your vector bundle which is Hermitian-Yang-Mills, i.e. it's curvature form is orthogonal to the Kahler form. Such a metric exists if and only if the bundle is a sum of slope stable bundles.

In general this can be seen as an infinite dimensional version of GIT vs symplectic reduction. The condition that a metric be HYM says that a certain moment map vanishes for this connection.

To start, I would recommend the following articles:

• "The Yang-Mills equations over Riemann surfaces" by Atiyah-Bott.

• "A new proof of a theorem of Narasimhan-Seshadri" by Donaldson.

• "Anti-self-dual Yang-Mills connections over complex algebraic surfaces" by Donaldson.

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As Charlie Frohman says, for curves this is the Narasimhan-Seshadri correspondonce. For Kahler manifolds of higher dimensions it is called the Hitchin-Kobayashi correspondence, proved by Donaldson-Uhlenbeck-Yau. You search for a Hermitian metric on your vector bundle which is Hermitian-Yang-Mills, i.e. it's curvature form is orthogonal to the Kahler form. Such a metric exists if and only if the bundle is a sum of slope stable bundles.

In general this can be seen as an infinite dimensional version of GIT vs symplectic reduction. The condition that a metric be HYM says that a certain moment map vanishes for this connection.