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$\pi : \hat{X} \to X$isacoveringmapand$dF=\pi^*$\mathbf{\xi }$$. We also discussed the criteria about when such a manifold $X$ admits a proper holomorphic mapping onto a Riemann surface.