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

TL;DR

This paper investigates the topological and mapping properties of weakly 1-complete manifolds with holomorphic one-forms, establishing conditions for connectivity and proper mappings onto Riemann surfaces.

Contribution

It provides new criteria for the existence of proper holomorphic maps from weakly 1-complete manifolds to Riemann surfaces based on properties of holomorphic one-forms.

Findings

Connectivity of the pair $(\hat{X}, F^{-1}(z))$ established

Criteria for proper holomorphic mappings onto Riemann surfaces derived

Conditions on holomorphic one-forms influencing manifold topology

Abstract

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.