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Holomorphic structures and connections on differentiable fibre bundles. (English) Zbl 0705.53030
One of the many important results of Atiyah-Hitchin-Singer [M. F. Atiyah, N. J. Hitchin and I. M. Singer, Proc. R. Soc. Lond., Ser. A 362, 425-461 (1978; Zbl 0389.53011)] (see also I. M. Singer [Pac. J. Math. 9, 585-590 (1959; Zbl 0086.151)], P. A. Griffiths [Am. J. Math. 88, 366-446 (1966; Zbl 0147.075)], M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond., A 308, 523-615 (1983; Zbl 0509.14014)]) tells that if \(\pi\) : \(E\to M\) is a \(C^{\infty}\) hermitian vector bundle with the complex basis N, then there is a natural bijection between the set of equivalence classes of holomorphic structures on E and the set of unitary connections of E whose curvature form has no term of the complex type (0,2) on M (i.e., the curvature has the complex type (1,1) on M) modulo gauge equivalence. Atiyah, Hitchin and Singer [loc. cit.] also notice that the result extends to principal bundles with a complex structure group G which has a compact real form. In the present note, we establish corresponding results for bundles with an arbitrary complex structure group (Theorems 1.5, 2.7). We give a rather detailed description of holomorphic structures on principal bundles, while correlating with R. S. Millman’s paper [Trans. Am. Math. Soc. 166, 71-99 (1972; Zbl 0214.219)]. We also obtain a generalization of a theorem of M. F. Atiyah [Trans. Am. Math. Soc. 85, 181-207 (1957; Zbl 0078.160)] concerning the vanishing of the characteristic classes of a bundle with holomorphic connection, and define secondary characteristic classes for these bundles if they are \(C^{\infty}\) trivial.

MSC:
53C56 Other complex differential geometry
32Q20 Kähler-Einstein manifolds
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References:
[1] M.F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207 · Zbl 0078.16002
[2] M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. London, A308 (1982), 523-615 · Zbl 0509.14014
[3] M.F. Atiyah, N.J. Hitchin and I.M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. London, A362 (1978), 425-461 · Zbl 0389.53011
[4] P.A. Griffiths, The extension problem in complex analysis II. Embeddings with positive normal bundle. Amer. J. Math. 88 (1966), 366-446 · Zbl 0147.07502
[5] M. Inoue, S. Kobayashi and T. Ochiai, Holomorphic affine connections on compact complex surfaces, J. Fac. Sci. Univ. Tokyo. 27 (1980), 247-264 · Zbl 0467.32014
[6] D.L. Johnson, Smooth moduli and secondary characteristic classes of analytic vector bundles. To appear
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[8] D. Lehmann, Classes caractéristiques exotiques et J-connexité des espaces de connexions, Ann. Inst. Fourier, Grenoble 24(3), (1974), 267-306 · Zbl 0268.57009
[9] R.S. Millman, Complex structures on real product bundles with applications to differential geometry, Trans. Amer. Math. Soc. 166 (1972), 71-99 · Zbl 0214.21901
[10] I.M. Singer, The geometric interpretation of a special connection, Pacific J. Math. 9(1959), 585-590 · Zbl 0086.15101
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