Torsion points of sections of lagrangian torus fibrations and the chow ring of Hyper-Kähler manifolds

Abstract

Let ϕ: X → B be a Lagrangian fibration on a projective irreducible hyper-Kähler manifold. Let M ∈Pic X be a line bundle whose restriction to the general fiber Xb of ϕ is topologically trivial. We prove that if the fibration is isotrivial or has maximal variation and X is of dimension ≤ 8, the set of points b such that the restriction M∣Xb is torsion is dense in B. We give an application to the Chow ring of X, providing further evidence for Beauville’s weak splitting conjecture.