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2012

Webs of Lagrangian Tori in Projective Symplectic Manifolds

Hwang, Jun-Muk, Weiss, Richard M.

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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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Wielandt, H.: Kriterien für Subnormalität in endlichen Gruppen. Math. Zeit. 138
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Fulton, W.: Intersection theory
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Guillemin, V., Sternberg, S.: Symplectic techniques in physics
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Voisin, C.: Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes. In Complex projective geometry (Trieste, 1989/Bergen, 1989)
1992
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Greb, D., Lehn, C. and Rollenske, S.: Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville. arXiv:1105.3410
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Hwang, J.-M.: Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. math. 174
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Campana F., Oguiso K. and Peternell, T.: Non-algebraic hyperkähler manifolds. J. Differential Geom. 85
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Beauville, A.: Holomorphic symplectic geometry: a problem list. In Complex and differential geometry
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