[Paper Review] Hypersurface Singularities and Milnor Equisingularity

Lê Dũng Tráng, David B. Massey|ArXiv.org|Apr 19, 2005

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes that the rank of the reduced homology group $\widetilde{H}_{n-s}(F_f)$ of the Milnor fiber at the origin completely determines whether a complex hypersurface singularity $f$ defines a simple $\mu$-constant family with constant Milnor number over a smooth base. The key result shows that equality between this Betti number and the generic $s$-dimensional Lê number $\lambda^s_f(\mathbf{0})$ holds if and only if the family is Milnor equisingular, linking topological invariants to equisingularity via perverse sheaves and vanishing cycles.

ABSTRACT

Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf 0}$, of $f$ at the origin, with either integral or $\mathbb Z/p\mathbb Z$ coefficients. If the critical locus of $f$ has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number of $F_{f, \mathbf 0}$ completely determines if $f$ defines a family of isolated singularities, over a smooth base, with constant Milnor number. This result has a nice interpretation in terms of the structure of the vanishing cycles as an object in the perverse category.

Motivation & Objective

• To determine when a hypersurface singularity $f$ defines a family of isolated singularities with constant Milnor number over a smooth base.
• To characterize Milnor equisingularity in terms of topological invariants of the Milnor fiber.
• To understand the structure of vanishing cycles in the perverse category and its relation to equisingularity.
• To establish effective, computable bounds on Betti numbers of the Milnor fiber for non-isolated singularities.
• To investigate when the vanishing cycles of the constant sheaf on affine space are semi-simple or decomposable.

Proposed method

• The authors use the Milnor fiber $F_f$ and compute its reduced homology groups $\widetilde{H}_k(F_f)$, focusing on the middle-dimensional group $\widetilde{H}_{n-s}(F_f)$.
• They employ the $s$-dimensional Lê number $\lambda^s_f(\mathbf{0})$, defined via local Milnor numbers along irreducible components of the critical locus $\Sigma f$.
• The analysis relies on the generic choice of coordinates $\mathbf{z} = (z_0, \dots, z_{s-1})$ to define normal slices and compute the $s$-dimensional Lê number.
• The paper uses the theory of perverse sheaves and the vanishing cycle functor $\phi_f[-1]\mathbb{Z}^\bullet_\mathcal{U}[n+1]$ to interpret equisingularity in derived categories.
• It applies the Sebastiani-Thom theorem and Cerf diagram techniques to analyze examples and test bounds.
• The proof of the main theorem reduces the general case to the $s=1$ case via induction and topological invariance.

Experimental results

Research questions

• RQ1 When does the rank of $\widetilde{H}_{n-s}(F_f)$ equal the generic $s$-dimensional Lê number $\lambda^s_f(\mathbf{0})$?
• RQ2 What is the precise topological condition that ensures a family of singularities has constant Milnor number?
• RQ3 Under what conditions is the vanishing cycle complex $\phi_f[-1]\mathbb{Z}^\bullet_\mathcal{U}[n+1]$ semi-simple as a perverse sheaf?
• RQ4 Can the Betti numbers of the Milnor fiber be effectively bounded in the non-isolated case?
• RQ5 Which perverse sheaves arise as vanishing cycles of the constant sheaf on $\mathbb{C}^{n+1}$?

Key findings

• The rank of $\widetilde{H}_{n-s}(F_f)$ equals $\lambda^s_f(\mathbf{0})$ if and only if $f$ defines a simple $\mu$-constant family at the origin.
• For $s=1$, the Milnor equisingularity condition is equivalent to $\operatorname{rank}\widetilde{H}_{n-1}(F_f) = \lambda^1_f(z_0)(\mathbf{0})$ or the same equality over $\mathbb{Z}/p\mathbb{Z}$ coefficients.
• If the $s$-dimensional components of $\Sigma f$ are smooth, then $\bigoplus_\nu (\mathbb{Z}^{\mu^\circ_\nu})^\bullet_\nu[s]$ is a direct summand of $\phi_f[-1]\mathbb{Z}^\bullet_\mathcal{U}[n+1]$ if and only if $f$ is Milnor equisingular.
• In the Milnor equisingular case, the vanishing cycle complex is isomorphic to $(\mathbb{Z}^{\mu^\circ_{\Sigma f}})^\bullet_{\Sigma f}[s]$.
• The vanishing cycle complex is not semi-simple in non-trivial cases, as shown by counterexamples where $\Sigma f$ is homeomorphic to $\mathbb{C}$ but the sheaf structure remembers the ambient hypersurface.
• Counterexamples exist showing that the Betti number can equal the sum of local Milnor numbers even when the family is not equisingular, indicating that stronger bounds require additional data like Cerf diagrams or monodromy.