Modification of Convex Ends to Cylindrical and Symplectic Foliations

TL;DR

This paper demonstrates how to modify symplectic structures on Milnor fibers of complex singularities to have cylindrical ends, enabling the construction of leafwise symplectic structures and closed symplectic 4-manifolds.

Contribution

It introduces a method to modify symplectic structures on Milnor fibers, facilitating new constructions in symplectic topology.

Findings

Symplectic structures on Milnor fibers can be modified to be cylindrical at the end.

The foliation on S^5 admits a leafwise symplectic structure.

Construction of certain closed symplectic 4-manifolds is achieved.

Abstract

We show that the natural symplectic structure on the Milnor fiber of an isolated singularity in complex three variables whose link fibers over the circle can be modified into one which is cylindrical at the end. As a consequence we see that the foliation of codimension one on S^5 which is adapted to the Milnor open book of S^5 associated with such a singularity admits a leafwise symplectic structure. The modification also enables us to construct certain closed symplectic 4-manifolds.