On Abel–Jacobi maps of Lagrangian families

Abstract

We study in this article the cohomological properties of Lagrangian families on projective hyper-Kähler 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 cohomology 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.