Kobayashi–Hitchin correspondence

In differential geometry, the Kobayashi–Hitchin correspondence(or Donaldson-Uhlenbeck-Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi Kobayashi and Nigel Hitchin, who independently conjectured in the 1980s that the moduli spaces of stable vector bundles and Einstein–Hermitian vector bundles over a complex manifold were essentially the same. This was proved by Donaldson for algebraic surfaces and later for algebraic manifolds, by Uhlenbeck and Yau for Kähler manifolds, and by Li and Yau for complex manifolds.

There is a folklore conjecture right after S.-T. Yau's proof of the Calabi conjecture that polystable bundles admit Hermitian Yang-Mills connection. This is partially due to the argument of Bogomolov and the success of Yau's work on constructing global geometric structures in Kahler geometry.

The most difficult part were accomplished by Simon Donaldson ^[1] for algebraic surfaces and Uhlenbeck-Yau for general case around 1982, announced in various seminars and appeared in print in 1985.^[2]

Soon after that, there are some formal publication of the conjecture due to Kobayashi.^[3] The program to carry out this deep theorem inspired by the work of Yau and Bogomolov is also called Donaldson-Uhlenbeck-Yau correspondence or DUY theorem. The proof of Uhlenbeck-Yau was the key to prove the later works in this direction, including the famous works of Carlos Simpson ^[4] on Higgs bundle. This later work is also called SUY theorem on Higgs bundle.

References

1. S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), 1--26.
2. K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles.Frontiers of the mathematical sciences: 1985 (New York, 1985). Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257--S293.
3. S. Kobayashi, Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A. Math. Sci., 58 (1982), 158--162.
4. C. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1 (1988), 867--918.

• Lübke, Martin; Teleman, Andrei (1995), The Kobayashi–Hitchin correspondence, River Edge, NJ: World Scientific Publishing Co. Inc., ISBN 9789810221683, MR 1370660
• Uhlenbeck, K.; Yau, Shing-Tung (1986), "On the existence of Hermitian–Yang–Mills connections in stable vector bundles", Communications on Pure and Applied Mathematics 39: S257–S293, doi:10.1002/cpa.3160390714, ISSN 0010-3640, MR 861491