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# Kobayashi–Hitchin correspondence

In differential geometry, the Kobayashi–Hitchin correspondence 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.

References

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

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