Do uniruled six-manifolds contain Sol Lagrangian submanifolds?

Details Do uniruled six-manifolds contain Sol Lagrangian submanifolds? Do uniruled six-manifolds contain Sol Lagrangian submanifolds?

2010-01-17

arxiv.org/abs/1001.2927v1

Mathematics

Symplectic Geometry

23 pages, 2 figures

Scientific paper

10.1093/imrn/rnr063

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. Koll\'ar. Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.

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