Tube classes over elementary vanishing cycles
Erjuan Fu

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 be a closed Riemann surface. When 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 .
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory · Commutative Algebra and Its Applications
