On Abel-Jacobi Maps of Lagrangian Families
Chenyu Bai

TL;DR
This paper investigates the conditions under which Abel-Jacobi maps of Lagrangian families on hyper-Kähler manifolds vanish or are non-trivial, providing criteria and examples that deepen understanding of their cohomological properties.
Contribution
It introduces a criterion for the vanishing of Abel-Jacobi maps in Lagrangian families and constructs examples demonstrating the criterion's optimality.
Findings
Maximal variation of Hodge structures implies trivial Abel-Jacobi map.
Constructed Lagrangian families with non-trivial Abel-Jacobi maps.
Provided a criterion for the vanishing of Abel-Jacobi maps.
Abstract
We study in this article the cohomological properties of Lagrangian families on projective hyper-K\"ahler manifolds. First, we give a criterion for the vanishing of Abel-Jacobi maps of Lagrangian families. Using this criterion, we show that under a natural condition, if the variation of Hodge structures on the degree cohomomology of the fibers of the Lagrangian family is maximal, its Abel-Jacobi map is trivial. We also construct Lagrangian families on generalized Kummer varieties whose Abel-Jacobi map is not trivial, showing that our criterion is optimal.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Geometric Analysis and Curvature Flows
