Lagrangian fibrations on hyperk\"ahler manifolds - On a question of Beauville

TL;DR

The paper proves that non-projective hyperk"ahler manifolds with a Lagrangian torus admit a Lagrangian fibration, and constructs almost holomorphic fibrations in the projective case under certain conditions, linking to minimal models.

Contribution

It establishes the existence of Lagrangian fibrations on non-projective hyperk"ahler manifolds and constructs almost holomorphic fibrations in the projective case with conditions, advancing understanding of their structure.

Findings

Non-projective hyperk"ahler manifolds with a Lagrangian torus admit a Lagrangian fibration.

In the projective case, almost holomorphic Lagrangian fibrations exist under specific conditions.

Existence of smooth minimal models for these fibrations, making them holomorphic on birational models.

Abstract

Let X be a compact hyperk\"ahler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair (X, L), which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperk\"ahler manifold birational to X on which the fibration is holomorphic.