On Abel-Jacobi Maps of Lagrangian Families

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 $1$ 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.