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Almost toric symplectic four-manifolds. (English) Zbl 1197.53103

Summary: Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that include both toric manifolds and the K3 surface. We classify closed almost toric four-manifolds up to diffeomorphism and indicate precisely the structure of all almost toric fibrations of closed symplectic four-manifolds. A key step in the proof is a geometric classification of the singular integral affine structures that can occur on the base of an almost toric fibration of a closed four-manifold. As a byproduct we provide a geometric explanation for why a generic Lagrangian fibration over the two-sphere must have 24 singular fibers.

MSC:

53D05 Symplectic manifolds (general theory)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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