nLab
Donaldson-Uhlenbeck-Yau theorem

Context

\infty-Chern-Weil theory

Differential cohomology

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 1J^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

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