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

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.

Tube classes over elementary vanishing cycles

“…Compare to the Strong Tube Theorem in [2], our result works for any complex projective curve which is not necessary a hyperplane section of a surface. The key point in our proof is the mapping class group action instead of the monodromy action in [2].…”

“…which is the topological Abel-Jacobi mapping [2,7]. Furthermore, in order to combine the topological Abel-Jacobi map a top X,t for all smooth hyperplane sections together, we consider the incidence variety…”