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Milnor Fiber Consistency via Flatness

Alex Hof

TL;DR

The paper tackles the problem of understanding how the $Milnor\ fibration$ behaves in holomorphic families by tying it to the flatness of the scheme-theoretic critical locus via normal cones and relative conormal spaces. It develops a Stratification Theorem that connects flatness to the existence of a Whitney stratification satisfying the $Thom\ (a_f)$ condition on strata, and a Fibration Theorem guaranteeing a smooth locally trivial fibration away from a discriminant, enabling Milnor-fibration information to be read from deformations. These results yield global consequences for homogeneous polynomials—partitioning their coefficient spaces into finitely many strata where the affine Milnor fibration type is constant—and extend to deformations where the critical locus is a complete intersection. The framework is compared to and sometimes strengthens prior approaches (e.g., finite determinacy, Lê numbers), with several explicit examples illustrating both the power and the limitations of the method in practice.

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.

Milnor Fiber Consistency via Flatness

TL;DR

The paper tackles the problem of understanding how the behaves in holomorphic families by tying it to the flatness of the scheme-theoretic critical locus via normal cones and relative conormal spaces. It develops a Stratification Theorem that connects flatness to the existence of a Whitney stratification satisfying the condition on strata, and a Fibration Theorem guaranteeing a smooth locally trivial fibration away from a discriminant, enabling Milnor-fibration information to be read from deformations. These results yield global consequences for homogeneous polynomials—partitioning their coefficient spaces into finitely many strata where the affine Milnor fibration type is constant—and extend to deformations where the critical locus is a complete intersection. The framework is compared to and sometimes strengthens prior approaches (e.g., finite determinacy, Lê numbers), with several explicit examples illustrating both the power and the limitations of the method in practice.

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.
Paper Structure (14 sections, 19 theorems, 16 equations, 1 figure)

This paper contains 14 sections, 19 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and References
  3. Complex-Analytic Geometry and Cones
  4. Stratifications and the Milnor Fibration
  5. Algebraic Lemmas
  6. Normal Cones and Infinitesimal Neighborhoods
  7. Relative Conormal Spaces
  8. Proof of the Stratification Theorem
  9. Proof of the Fibration Theorem
  10. Global Consequences
  11. Complete Intersections
  12. Comparison with Existing Techniques
  13. Examples
  14. Bibliography

Key Result

Theorem 1.1

Let $u \ge 0$ be an integer, $X'$ a complex-analytic manifold, $X \subseteq X' \times \mathbb{C}^u$ an open subset, $\pi: X \to \mathbb{C}^u$ the restriction to $X$ of the projection onto $\mathbb{C}^u$, and $F: X \to \mathbb{C}$ a holomorphic function which is not constant on any connected componen

Figures (1)

  • Figure 1: The critical loci of Example \ref{['ex:wup']} over $t = 0$ and $t \ne 0$ respectively ($s = 5t^2$)

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: e.g., teissier_hunt
  • Example 2.2
  • Proposition 2.3: am_cones
  • Definition 2.4
  • Proposition 2.5: rees
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • ...and 32 more