Milnor Fiber Consistency via Flatness

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.