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Homogeneous vector bundles and stability. (English) Zbl 0585.32031

If E is a homogeneous vector bundle over a homogeneous algebraic manifold \(M=G/G_ 0\) of a compact Lie group G and if the isotropy subgroup \(G_ 0\) is irreducible on the fibre \(E_ 0\) of E at the origin \(0\in M\), then E with any G-invariant Hermitian structure h is an Einstein-Hermitian vector bundle (in the sense of the author, Nagoya Math. J. 77, 5-11 (1980; Zbl 0432.53049), and Proc. Japan Acad., 158-162 (1982; Zbl 0538.32021)]. From a differential geometric criterion for stability [in the second paper cited above] it follows that E is H-semistable for any ample line bundle H. In order to see whether E is indeed H-stable or not, the holonomy group of E is studied. Next the author gives a differential geometric proof to the theorem of S. Ramanan [Topology 5, 159-177 (1966; Zbl 0138.186)] and H. Umemura [Nagoya Math. J. 69, 131-138 (1978; Zbl 0345.14017)] that every irreducible homogeneous vector bundle over a Kähler C-space M (of H. C. Wang) is H-stable for any ample line bundle H. An application for the null correlation bundles over \(P_{2n+1}\) are given. Using complex contact structures, an example of stable Einstein-Hermitian bundle is constructed.
Reviewer: Z.Olszak

MSC:

32L05 Holomorphic bundles and generalizations
32M10 Homogeneous complex manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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[1] DOI: 10.1007/BF01169586 · Zbl 0558.53037 · doi:10.1007/BF01169586
[2] Progress in Math. 3 (1980)
[3] Mathematics Seminar Notes 41 (1982)
[4] DOI: 10.3792/pjaa.58.158 · doi:10.3792/pjaa.58.158
[5] Nagoya Math. J. 77 pp 5– (1980) · Zbl 0432.53049 · doi:10.1017/S0027763000018602
[6] DOI: 10.1090/S0002-9939-1959-0111061-8 · doi:10.1090/S0002-9939-1959-0111061-8
[7] DOI: 10.1073/pnas.40.12.1147 · Zbl 0058.16002 · doi:10.1073/pnas.40.12.1147
[8] J. spaces, Math. Mech. 14 pp 1033– (1965)
[9] DOI: 10.1090/S0002-9939-1962-0137126-2 · doi:10.1090/S0002-9939-1962-0137126-2
[10] DOI: 10.2307/2372397 · Zbl 0055.16603 · doi:10.2307/2372397
[11] DOI: 10.1090/pspum/003/0124863 · doi:10.1090/pspum/003/0124863
[12] Nagoya Math. J. 69 pp 131– (1978) · Zbl 0345.14017 · doi:10.1017/S0027763000017992
[13] Symposia Math 26 pp 139– (1981)
[14] DOI: 10.1016/0040-9383(66)90017-6 · Zbl 0138.18602 · doi:10.1016/0040-9383(66)90017-6
[15] (1982)
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