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Stability of Einstein-Hermitian vector bundles. (English) Zbl 0558.53037
On a compact Kähler manifold there is a close relationship between stable bundles in the sense of Mumford-Takemoto and the Einstein- Hermitian vector bundles of S. Kobayashi [Einstein-Hermitian vector bundles and stability, Proc. Sympos. Global Riemannian Geometry, Durham, England 1982]. These Einstein-Hermitian bundles are Hermitian vector bundles with a certain restriction on their curvature.
In this paper the author proves a theorem announced by Kobayashi: An Einstein-Hermitian bundle on a compact Kähler manifold is semi-stable and a direct sum of stable Einstein-Hermitian line bundles. The proof is short and efficient.
Reviewer: M.G.Eastwood

53C55 Global differential geometry of Hermitian and Kählerian manifolds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Full Text: DOI EuDML
[1] Bogomolov,F.A.: Holomorphic tensors and vector bundles on projective varieties. Math.USSR Izvestija 13, 499-555 (1979) · Zbl 0439.14002 · doi:10.1070/IM1979v013n03ABEH002076
[2] Griffiths,P.A.: The extension problem in complex analysis II. Amer.J.Math. 88, 366-446 (1966) · Zbl 0147.07502 · doi:10.2307/2373200
[3] Griffiths,P.A., Harris,J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001
[4] Kobayashi,S.: Curvature and stability of vector bundles. Proc. Japan Acad. 58 A 4, 158-162 (1982) · Zbl 0538.32021
[5] Kobayashi,S.: Einstein-Hermitian vector bundles and stability. to appear in Proc.Symp. Global Riemannian Geometry, Durham, England (1982) · Zbl 0546.53041
[6] Kobayashi,S.: First Chern class and holomorphic tensor fields. Nagoya Math.J. 77, 5-11 (1980) · Zbl 0432.53049
[7] Lübke,M.: Chernklassen von Hermite-Einstein-Vektorbündeln, Math.Ann. 260, 133-141 (1982) · Zbl 0481.53058 · doi:10.1007/BF01475761
[8] Lübke,M.: Hermite-Einstein-Vektorbündel. Dissertation, Bayreuth 1982. · Zbl 0569.53030
[9] Okonek,C., Schneider,M., Spindler,H.: Vector bundles on complex projective spaces. Boston-Basel-Stuttgart: Birkhäuser 1980 · Zbl 0438.32016
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