Tube classes over elementary vanishing cycles

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 $X$.

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