Milnor Fiber Consistency via Flatness

TL;DR

This paper introduces a new algebraic approach to understanding Milnor fibrations, providing criteria for deformations of holomorphic functions to preserve stratifications and fibration structures, with applications to polynomial spaces.

Contribution

It offers novel criteria for deformations to admit compatible stratifications and Milnor fibrations, advancing the algebraic understanding of Milnor fiber consistency.

Findings

Partitioned polynomial space into finitely many subsets with constant Milnor fiber type

Established criteria for deformations with complete intersection critical loci to be well-behaved

Provided a new algebraic perspective on Milnor fibrations and stratifications

Abstract

We describe a new algebro-geometric perspective on the study of the Milnor fibration and, as a first step toward putting it into practice, prove powerful criteria for a deformation of a holomorphic function germ to admit a stratification on its domain partially satisfying the Thom condition and, more generally, to respect the Milnor fibration of the original germ in an appropriate sense. As corollaries, we obtain a method of partitioning the space of homogeneous polynomials of a fixed degree into finitely many locally closed subsets such that the fiber diffeomorphism type of the Milnor fibration is constant along each subset and a criterion under which deformations of a function with critical locus a complete intersection will be well-behaved.