A new approach towards Lefschetz $(1, 1)$-Theorem

TL;DR

This paper introduces a novel geometric approach to Lefschetz's (1,1)-Theorem using two-parameter families and topological Abel--Jacobi mappings, providing insights for higher dimensions.

Contribution

It constructs new geometric methods involving two-parameter families and generalizes existing theorems like Schnell's tube theorem to integral homology groups.

Findings

Generalized Schnell's tube theorem to integral homology groups

Proved a Jacobi-type inversion theorem

Provided a geometric description of deformation spaces of vanishing cycles

Abstract

Let $S$ be a complex projective surface. Lefschetz originally proved Lefschetz $(1, 1)$--Theorem by studying a Lefschetz pencil of hyperplane sections of $S$ and the Abel--Jacobi mapping. In this paper, we attack Lefschetz $(1, 1)$--Theorem by constructing certain two-parameter families of twice hyperplane sections of $S$ and then applying the topological Abel--Jacobi mapping. Our geometric constructions would give an inductive approach and some insight for higher dimensional cases. We prove a strong tube theorem which generalizes Schnell's tube theorem to integral homology groups for complex projective curves and then obtain a Jacobi-type inversion theorem. In the end, we give a geometric description for the deformation space of an elementary vanishing cycle over a generic net.