Kobayashi–Hitchin correspondence

In differential geometry and gauge theory, 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 Simon Donaldson for algebraic surfaces and later for algebraic manifolds, by Karen Uhlenbeck and Shing-Tung Yau for Kähler manifolds, and by Jun Li and Yau for complex manifolds.

Overview

There was folklore conjecture right after Yau's proof of the Calabi conjecture that polystable bundles admit Hermitian Yang-Mills connections. This is partially due to the argument of Fedor Bogomolov and the success of Yau's work on constructing global geometric structures in Kähler geometry.

The most difficult part was accomplished by 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 Shoshichi 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 further advances in this direction, including the famous result of Carlos Simpson[4] on Higgs bundles. This result is also called the SUY theorem on Higgs bundles.

Notes
  1. Simon K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proceedings of the London Mathematical Society (3) 50 (1985), 1-26.
  2. Karen Uhlenbeck and Shing-Tung Yau, On the existence of Hermitian–Yang-Mills connections in stable vector bundles.Frontiers of the mathematical sciences: 1985 (New York, 1985). Communications on Pure and Applied Mathematics 39 (1986), no. S, suppl., S257-S293.
  3. Shoshichi Kobayashi, Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A. Math. Sci., 58 (1982), 158-162.
  4. Carlos Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, Journal of the American Mathematical Society 1 (1988), 867–918.

References
  • Lübke, Martin; Teleman, Andrei (1995), The Kobayashi–Hitchin correspondence, River Edge, NJ: World Scientific Publishing Co. Inc., ISBN 9789810221683, MR 1370660
  • Uhlenbeck, Karen; 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 0861491
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