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Do uniruled six-manifolds contain Sol Lagrangian submanifolds?

Abstract : We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Kollár. Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.
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Submitted on : Sunday, January 17, 2010 - 8:33:43 PM
Last modification on : Friday, January 7, 2022 - 2:14:03 PM
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Frédéric Mangolte, Jean-yves Welschinger. Do uniruled six-manifolds contain Sol Lagrangian submanifolds?. International Mathematics Research Notices, Oxford University Press (OUP), 2012, 2012, pp.1569-1602. ⟨10.1093/imrn/rnr063⟩. ⟨hal-00447962⟩



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