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

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