| The problems presented here generally relate to a discussion of the conjecture of Shafarevich [5] that the universal covering space of a connected smooth complex projective variety is holomorphically convex. There have been many partial results concerning this problem, and no known counterexamples at present. In particular Napier proved a partial result for a space of dimension 2 ([3]). We hope to extend this to a space of dimension n. The main result here is a generalization of the first step of his method.;Theorem. Let X be a complex manifold of dimension n, C a compact analytic subset of X of dimension at most q, and pi : X˜ → X a covering space of X. Assume that C admits a projective embedding and pi -1(C) has no compact irreducible components. Then there is a Cinfinity exhaustion function on X˜ whose restriction to a uniform neighborhood of pi-1(C) is strongly q-convex. |
