A remark on a question of Beauville about lagrangian fibrations

TL;DR

This paper proves that any lagrangian torus on a hyperkähler fourfold is necessarily a fiber of an almost holomorphic lagrangian fibration, confirming a specific case of Beauville's question.

Contribution

It establishes that lagrangian tori in hyperkähler fourfolds always serve as fibers of an almost holomorphic lagrangian fibration, addressing Beauville's question.

Findings

Lagrangian tori are fibers of almost holomorphic fibrations.

The result applies specifically to hyperkähler fourfolds.

Supports Beauville's conjecture in this context.

Abstract

This note is a proof of the fact that a lagrangian torus on a hyperkaehler fourfold is always a fiber of an almost holomorphic lagrangian fibration.