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

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