Complex analytic connections in fibre bundles

Abstract

Introduction. In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles. The ideas outlined above are developed in considerable detail, and they are applied in particular to a problem first studied by Weil [17], namely the problem of characterizing those fibre bundles which arise from a representation of the fundamental group. We show how Weil's main result fits into the general picture, and we discuss various aspects of the problem. As no complete exposition of the theory of complex analytic fibre bundles has as yet been published, this paper should start with a basic exposition of this nature. However this would be a major undertaking in itself, and instead we shall simply summarize in ?1 the terminology and results on vector bundles which we require, and for the rest we refer to Grothendieck [8], Serre [12], and Hirzebruch [9].