S1-invariant symplectic hypersurfaces in dimension 6 and the Fano condition

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We prove that any symplectic Fano 6-manifold M with a Hamiltonian S1-
action is simply connected and satises c1c2(M) = 24. This is done by showing
that the xed submanifold Mmin M on which the Hamiltonian attains its minimum is dieomorphic to either a del Pezzo surface, a 2-sphere or a point. In the case when dim(Mmin) = 4, we use the fact that symplectic Fano 4-manifolds are symplectomorphic to del Pezzo surfaces. The case when dim(Mmin) = 2 involves a study of 6-dimensional Hamiltonian S1-manifolds with Mmin dieomorphic to a surface of positive genus. By exploiting an analogy with the algebro-geometric situation we construct in each such 6-manifold an S1-invariant symplectic hypersurface F(M) playing the role of a smooth bre of a hypothetical Mori bration over Mmin. This relies upon applying Seiberg-Witten theory to the resolution of symplectic 4-orbifolds occurring as the reduced spaces of M.
Original languageEnglish
Pages (from-to)221–285
Number of pages65
JournalJournal of Topology
Issue number1
Early online date12 Jan 2019
Publication statusPublished - Mar 2019


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