Torsion points of sections of Lagrangian torus fibrations and the Chow ring of hyper-K\"ahler manifolds

TL;DR

This paper investigates the distribution of torsion points of line bundles on Lagrangian fibrations of hyper-K"ahler manifolds, revealing density results and implications for the Chow ring structure.

Contribution

It establishes density of torsion restrictions for line bundles on hyper-K"ahler fibrations with maximal variation or isotriviality, and applies these results to the Chow ring.

Findings

Torsion points are dense in the base for certain fibrations.

Results apply to hyper-K"ahler manifolds of dimension up to 8.

Provides a model using elliptic fibrations for the main argument.

Abstract

Let $\phi:X\rightarrow B$ be a Lagrangian fibration on a projective irreducible hyper-K\"ahler manifold of dimension $\leq8$. Let $M\in {\rm Pic}\,X$ be a line bundle whose restriction to the general fiber $X_b$ of $\phi$ is topologically trivial. We prove that if the fibration has maximal variation or is isotrivial, the set of points $b$ such that the restriction $M_{\mid X_b}$ is torsion is dense in $B$. We give an application to the Chow ring of $X$. We prove a similar result for elliptic fibrations which gives a toy model for the argument.