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

TL;DR

This paper constructs genus-one Lefschetz fibrations on Milnor fibers of cusp and simple elliptic singularities, linking them to Lawson foliations on spheres and demonstrating their symplectic structures and gluing properties.

Contribution

It introduces explicit Lefschetz fibration structures on Milnor fibers of cusp and elliptic singularities and explores their topological and symplectic implications.

Findings

Milnor fibers admit $S^1$-parametric genus-one Lefschetz fibrations.

Lawson foliations on $S^5$ are pullbacks of Reeb foliations on $S^3$.

Gluing pairs of Milnor fibers yields K3 surfaces.

Abstract

We show that the total space of the Milnor fibration associated with any cusp or simple elliptic singularity in complex three variables admits an $S^1$-parametric genus-one Lefschetz fibration structure over the $2$-disk. As a consequence, we demonstrate that the Lawson type foliations on $S^5$ associated with such singularities can be regarded as the pullback of the Reeb foliation on $S^3$. This enables us to provide an alternative proof of a previous result by the third author, which states that every Lawson type foliation admits a leafwise symplectic structure. Also we see that a pair of such Milnor fibers can be glued together along boundary into a closed oriented 4-manifold exactly when the pair corresponds to one of the ten extended strange duality pairs among the cusp singularities. This gluing is compatible with the Lefschetz fibrations and the resultant 4-manifold is diffeomrphic to a K3 surface.