Lefschetz theorem for holomorphic one-forms on weakly 1-complete manifolds

Abstract

For a holomorphic one-form \({\xi }\) on a weakly 1-complete manifold X with certain properties, we will discuss the connectivity of the pair \((\hat{X},F^{-1}(z))\), where \(\pi :\hat{X} \rightarrow X\) is a covering map and F is a holomorphic function on \(\hat{X}\) such that \(dF=\pi ^*{\xi }\). We will also discuss the criteria about when such a manifold X admits a proper holomorphic mapping onto a Riemann surface.