# zbMATH — the first resource for mathematics

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
Full Text:
##### 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 [7] S. Kobayashi and K. Nomizu, Foundations of differential geometry I, II, Intersci. Publ., New York 1963, 1969 · Zbl 0119.37502 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.