On families of lagrangian tori on hyperkaehler manifolds

TL;DR

This paper discusses whether lagrangian tori on hyperkaehler manifolds are fibers of lagrangian fibrations, offering a new short proof for the non-algebraic case and insights for the algebraic case.

Contribution

It provides a concise alternative proof for Beauville's problem in the non-algebraic case and suggests a new approach for the algebraic case.

Findings

Confirmed lagrangian tori are fibers in the non-algebraic case.

Proposed a different approach for the algebraic case.

Provided a short, elegant proof for a known problem.

Abstract

This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different, very short solution in the non-algebraic case and make some observations suggesting a different approach in the algebraic case.