Webs of Lagrangian Tori in Projective Symplectic Manifolds

TL;DR

This paper proves that in certain symplectic manifolds, Lagrangian tori are fibers of special fibrations, using integrable systems and group theory, advancing understanding of hyperkähler geometry.

Contribution

It establishes that Lagrangian tori in projective symplectic manifolds are fibers of almost holomorphic Lagrangian fibrations, answering a question posed by Beauville.

Findings

Lagrangian tori in simply-connected projective symplectic manifolds have associated hypersurfaces disjoint from their deformations.

In compact hyperkähler manifolds, Lagrangian tori are fibers of almost holomorphic Lagrangian fibrations.

The proof combines action-angle variables and Wielandt's theory of subnormal subgroups.

Abstract

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville's. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt's theory of subnormal subgroups.