Webs of Lagrangian Tori in Projective Symplectic Manifolds

Details Webs of Lagrangian Tori in Projective Symplectic Manifolds Webs of Lagrangian Tori in Projective Symplectic Manifolds

2012-01-11

arxiv.org/abs/1201.2369v2

Mathematics

Algebraic Geometry

18 pages, minor latex problem fixed

Scientific paper

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.

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