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

Leung, Naichung Conan; Symington, Margaret

J. Symplectic Geom. 8, No. 2, 143-187 (2010).

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

Keywords:

K3 surfaces; closed symplectic four-manifolds; Lagrangian fibration; two-sphere