Kobayashi–Hitchin correspondence

Here we give the statement of the Kobayashi–Hitchin correspondence for arbitrary compact complex manifolds, a case where the above definitions of stability and special metrics can be readily extended.

The Kobayashi–Hitchin correspondence was one of the first instances of a general principle that has come to dominate geometry research since its proof: . Many results have been proven either as extensions or variations of the Kobayashi–Hitchin correspondence, or by direct analogy with the correspondence to seemingly disparate parts of geometry, and all of these results follow along this same principle. Here a summary of these generalisations or related results is given:

extremal objects in differential geometry correspond to stable objects in algebraic geometry