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 hyperkähler 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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Acknowledgements
When we first started working on Beauville’s question (Theorem 1.2), it was Keiji Oguiso who told us that it could be reduced to proving Theorem 1.1. We would like to thank him for this information and encouragement. This paper is dedicated to the memory of Professor Hyo Chul Myung. It was through him that the two authors became acquainted with each other five years ago. We believe that he would have been delighted to see our collaboration and regret that it had started after he passed away.
Jun-Muk Hwang is supported by National Researcher Program 2010-0020413 of NRF and MEST, and Richard Weiss is partially supported by DFG Grant KR 1669/7-1 and NSA Grant H98230-12-1-0230.
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Dedicated to the memory of Professor Hyo Chul Myung
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Hwang, JM., Weiss, R.M. Webs of Lagrangian tori in projective symplectic manifolds. Invent. math. 192, 83–109 (2013). https://doi.org/10.1007/s00222-012-0407-2
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DOI: https://doi.org/10.1007/s00222-012-0407-2