nLab
Donaldson-Uhlenbeck-Yau theorem
Context
$\infty$ -Chern-Weil theory
Differential cohomology
differential cohomology

Ingredients
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Fiber integration
Application to gauge theory
Contents
Idea
A series of results on moduli spaces of connections in the presence of complex structure due to (Donaldson 85 , Uhlenbeck-Yau 86 ), the most famous of which says that on a compact Kähler manifold any semistable holomorphic vector bundle with trivial determinant line bundle admits a Hermite-Einstein connection .

The theorem is recalled for instance as (Scheinost-Schottenloher 96, theorem 1.15 ). This result is a key step in the construction of the Kähler polarization on a moduli space of flat connections via symplectic reduction (Scheinost-Schottenloher 96, corollary 1.16 ), a non-abelian version of the Griffiths intermediate Jacobian $J^1$ (see there at Examples – Picard variety ).

The correspondence between semi-stable vector bundles and Hermite-Einstein connections holds more generally over complex manifolds , where it is known as the Kobayashi-Hitchin correspondence . The special case of that over Riemann surfaces in turn is essentially the Narasimhan-Seshadri theorem .

References
The original articles are

Simon Donaldson , Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles , Proc. LMS 50 (1985) 1-26

Karen Uhlenbeck , Shing-Tung Yau , On the existence of Hermitean Yang-Mills-connections on stable bundles over Kähler manifolds , Comm. Pure Appl. Math. 39 (1986) 257-293

Reviews in the context of discussion of Kähler polarization of moduli spaces of connections is in

Peter Scheinost, Martin Schottenloher , Metaplectic quantization of the moduli spaces of flat and parabolic bundles , J. reine angew. Mathematik, 466 (1996) (web )