Although our primary interest lies in holomorphic vector bundles, we begin this chapter with the study of connections in differentiable complex vector bundles. In order to discuss moduli of holomorphic vector bundles, it is essential to start with differentiable complex vector bundles. In discussing Chern classes it is also necessary to consider the category of differentiable complex vector bundles rather than the category of holomorphic vector bundles which is too small and too rigid.Most of the results in this chapter are fairly standard and should be well known to geometers. They form a basis for the subsequent chapters. As general references on connections, we mention Kobayashi-Nomizu [75] and Chern [22]. Connections in complex vector bundles (over real manifolds)Let M be an n-dimensional real C ∞ manifold and E a C ∞ complex vector bundle of rank (= fibre dimension) r over M . We make use of the following notations:A p = the space ofLet s = (s 1 , • • • , s r ) be a local frame field of E over an open set U ⊂ M , i.e., CONNECTIONS IN COMPLEX VECTOR BUNDLES (OVER COMPLEX MANIFOLDS)7provided with a flat structure. A connection D in E (i.e., a connection in P ) is said to be projectively flat if the induced connection inP is flat. As a special case of (1.2.6), we have

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