nLab
Donaldson-Uhlenbeck-Yau theorem

Contents

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

  • 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)

Last revised on July 15, 2014 at 02:07:00. See the history of this page for a list of all contributions to it.