Lagrangian fibrations on hyperk\"ahler fourfolds

TL;DR

This paper investigates Lagrangian fibrations on hyperk"ahler fourfolds, proving the existence of holomorphic fibrations with specific fibers and addressing a question posed by Beauville.

Contribution

It establishes that in four-dimensional hyperk"ahler manifolds with a Lagrangian subtorus, a holomorphic Lagrangian fibration always exists, confirming a special case of Beauville's question.

Findings

Existence of holomorphic Lagrangian fibrations in dimension four

Characterization of L-reduction and its relation to fibrations

Answer to Beauville's question in the fourfold case

Abstract

Let X be a projective hyperk\"ahler manifold containing a Lagrangian subtorus L. We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an almost holomorphic Lagrangian fibration with (strong) fibre L. In dimension four we prove that in the above situation there is always a holomorphic Lagrangian fibration with fibre L, thus answering a question of Beauville in this particular case.