Lefschetz fibrations on the Milnor fibers of cusp singularities and applications

We introduce Lefschetz fibration structures on the Milnor fibers of simple-elliptic and cusp singularities in complex three variables, whose regular fibers are diffeomorphic to the 2-torus. We know two ways to construct them and explain h-principle flavors in each construction. There are two major applications. One is a construction of leafwise symplectic structure on the associated codimension one foliations with the Milnor open books of the 5-sphere. 

The other is the topological decompositions of the K3 surface into two Milnor fibers according to the extended strange duality pairs. This further explains Pinkham's interpretation of the strange duality in terms of K3 lattice.  

This talk is based on the recent joint work with N. Kasuya, H. Kodama, and A. Mori.