The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof

TL;DR

This paper reviews the Milnor fiber conjecture for splice type surface singularities, providing an outline of its proof using tropical and log geometry techniques, especially the operation of rounding of complex logarithmic spaces.

Contribution

It offers a comprehensive overview and detailed outline of the proof of the Milnor fiber conjecture, integrating tropical and log geometry methods.

Findings

The conjecture relates splice diagram edges to Milnor fiber decompositions.

The proof employs tropical geometry and log geometry, notably the rounding of complex logarithmic spaces.

The approach generalizes classical techniques like real oriented blowup in Milnor fibration studies.

Abstract

Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are determined, up to a form of equisingularity, by decorated trees called splice diagrams. In 2005, Neumann and Wahl formulated their Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a special kind of decomposition into pieces of the Milnor fibers of the associated singularities. These pieces are constructed from the Milnor fibers of the splice type singularities determined by the subdiagrams on both sides of the chosen edge. In this paper we give an overview of this conjecture and a detailed outline of its proof, based on techniques from tropical geometry and log geometry in the sense of Fontaine and Illusie. The crucial log geometric ingredient is the operation of rounding of a complex logarithmic space introduced in 1999 by Kato and Nakayama. It is a functorial generalization of the operation of real oriented blowup. The use of the latter to study Milnor fibrations was pioneered by A'Campo in 1975.