Topological Abel-Jacobi Map and Mixed Hodge Structures
Yilong Zhang
TL;DR
This work resolves a question of whether two a priori different topological Abel-Jacobi constructions agree on vanishing cycles. By embedding the comparison in the framework of mixed Hodge structures and using Deligne’s $R$-split theory together with Lefschetz duality, the authors show that Zhao’s map from vanishing cohomology to the primitive intermediate Jacobian coincides with Schnell’s real-split construction, and provide an explicit residue-based pairing formula that identifies the two maps up to periods. The result unifies the topological and Hodge-theoretic perspectives on Abel-Jacobi theory in the relative setting $(X,Y)$ and its open complement $U=Xackslash Y$, confirming predictions about holomorphic variation along Hodge loci and enriching the Carlson–Griffiths interpretation in higher dimensions. This equivalence enhances robustness of the Abel-Jacobi formalism in families and offers a canonical framework for extending to more general extensions of real-split mixed Hodge structures, with potential applications to curvature and period computations in algebraic geometry.
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.