Topological Abel-Jacobi Map and Mixed Hodge Structures

TL;DR

This paper demonstrates the equivalence of two different constructions of the topological Abel-Jacobi map for complex projective varieties, linking geometric and Hodge-theoretic perspectives.

Contribution

It proves that Zhao's topological Abel-Jacobi map and Schnell's mixed Hodge structure-based map are the same, resolving a previously open question.

Findings

The two definitions of the Abel-Jacobi map coincide.

The result confirms the compatibility of geometric and Hodge-theoretic approaches.

Provides a unified understanding of the Abel-Jacobi map for complex varieties.

Abstract

For a smooth projective variety X of dimension 2n-1 over complex field, Zhao defined the topological Abel-Jacobi map, which sends vanishing cycles on a smooth hyperplane section Y to the middle dimensional primitive intermediate Jacobian of X. It agrees with Griffiths' Abel-Jacobi map on vanishing cycles that are algebraic and varies holomorphically on the locus of Hodge classes as hyperplane section deforms. On the other hand, Schnell proposed an alternative construction using the real-splitting property of the mixed Hodge structure on H^{2n-1}(X\Y). We show that the two definitions coincide, which answers a question of Schnell.