Parabolic exhaustions and analytic coverings (Q1323410)

Let \(\tau\) be a parabolic exhaustion on a Stein manifold \(X\) such that \(\tau\) is strictly plurisubharmonic at its zeros. The metric defined by \(\tau\) on the complement of its degeneracy locus \(D\) is shown to be flat if \(\tau\) is real-analytic or if ``most'' leaves of the associated Monge- Ampère foliation \({\mathcal F}\) abut the zeros of \(\tau\). Then, by an analysis of the singularities of \(\tau\), we show that the tangent bundle of \(X \backslash D\) extends to a flat hermitian bundle on \(X\) with a holomorphic section \(s\) such that \(\tau=\| s |^ 2\), and that \({\mathcal F}\) extends to a singular holomorphic foliation of \(X\). Also, \(\tau\) is the length-squared of an analytic covering of \(X\) onto a ball if and only if the monodromy of the \(\tau\)-connection is trivial. We obtain a characterization of affine algebraic manifolds as those \(X\) possessing \(\tau\) with finite monodromy and affine leaves.