Almost toric fibrations on K3 surfaces

It is known that a manifold diffeomorphic to a K3 surface admits an almost toric fibration (ATF). However, given a specific symplectic form on a K3 it is unclear if it admits an almost toric fibration with lagrangian fibres for the given form. In this talk, we prove that when a Kähler K3 surface admits a Type III Kulikov degeneration with a symplectic form taming the complex structure, the symplectic form admits an ATF whose base is the intersection complex of the degenerate fibre. Furthermore, we shall show that a smooth anti-canonical hypersurface in a smooth toric Fano threefold, equipped with a toric Kähler form, admits such a symplectic Kulikov model.

This is based on joint work with Yoel Groman.