Tube classes over elementary vanishing cycles

TL;DR

This paper extends Schnell's tube mapping from rational to integral homology for Riemann surfaces, relating it to the Abel-Jacobi map and showing that tube classes from elementary vanishing cycles form a large subgroup of the homology.

Contribution

It generalizes the tube mapping to integral homology and establishes its relation to the Abel-Jacobi map, demonstrating the subgroup's cofinite nature.

Findings

Tube classes from elementary vanishing cycles form a cofinite subgroup of the first integral homology.

The relation between tube mapping and the Abel-Jacobi map is established for integral homology.

The approach uses the mapping class group action to analyze the subgroup structure.

Abstract

Let $X$ be a closed Riemann surface. When $X$ is embedded into a projective space, the first rational cohomology group can be concretely obtained from the monodromy in the family of its smooth hyperplane sections by C. Schnell's tube mapping. We generalize this result to the first integral homology group by relating the tube mapping with the topological Abel--Jacobi mapping. By making use of the mapping class group action, we prove that all tube classes constructed from the elementary vanishing cycles form a cofinite subgroup of the first integral homology group of $X$.