Lefschetz fibrations on the Milnor fibers of cusp and simple elliptic singularities

Abstract: We show that there exists a Lefschetz fibration with the regular fiber diffeomorphic to $T^{2}$ on the Milnor fiber of any of cusp or simple elliptic singularities in complex three variables. As a consequence, we obtain a smooth topological decomposition of a K3 surface into two Milnor fibers according to each of the extended strange duality pairs between cusp singularities, so that an elliptic fibration structure of the K3 surface restricts to the Lefschetz fibration structure of each Milnor fiber.