Holomorphic structures and connections on differentiable fibre bundles (Q917950)

One of the many important results of Atiyah-Hitchin-Singer [\textit{M. F. Atiyah}, \textit{N. J. Hitchin} and \textit{I. M. Singer}, Proc. R. Soc. Lond., Ser. A 362, 425-461 (1978; Zbl 0389.53011)] (see also \textit{I. M. Singer} [Pac. J. Math. 9, 585-590 (1959; Zbl 0086.151)], \textit{P. A. Griffiths} [Am. J. Math. 88, 366-446 (1966; Zbl 0147.075)], \textit{M. F. Atiyah} and \textit{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 \textit{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 \textit{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.