Lé-Greuel Type Formulas for Functions with Isolated Singularities on Stratified Spaces

Core Concepts

This article presents a generalization of the classical Lé-Greuel formula for computing Milnor numbers of isolated complete intersection singularities. The generalized formulas apply to holomorphic functions with isolated singularities on stratified complex analytic spaces and express the Milnor numbers of these singularities in terms of holomorphic Euler characteristics of certain complexes.

Abstract

• Bibliographic Information: Zach, M. (2024). Some Lê-Greuel type formulae on stratified spaces. arXiv preprint arXiv:2411.02682v1.
• Research Objective: To generalize the classical Lé-Greuel formula for computing Milnor numbers of isolated complete intersection singularities to the broader context of holomorphic functions with isolated singularities on stratified complex analytic spaces.
• Methodology: The paper utilizes tools from stratified Morse theory, including Tib˘ar's Bouquet Theorem and the concept of relative polar varieties, to analyze the topology of Milnor fibers. It also employs sheaf-theoretic techniques, particularly the Koszul and Eagon-Northcott complexes, to express the Milnor numbers as holomorphic Euler characteristics.
• Key Findings: The paper establishes several formulas for computing the Milnor numbers µ(α;f) associated with each stratum V α of a stratified space X, where f is a holomorphic function with an isolated singularity on X. These formulas involve the holomorphic Euler characteristics of complexes constructed using the Koszul and Eagon-Northcott complexes applied to the pullbacks of the component functions of f and generic linear forms to the Nash blow-up of the stratum closures.
• Main Conclusions: The generalized Lé-Greuel type formulas provide a powerful tool for studying the topology of singularities in stratified spaces. They offer a way to compute Milnor numbers, which are important invariants of singularities, in terms of computable algebraic objects.
• Significance: This research significantly contributes to the field of singularity theory by extending a classical result to a more general setting. The generalized formulas have potential applications in various areas of mathematics, including algebraic geometry, topology, and differential equations.
• Limitations and Future Research: The paper primarily focuses on isolated singularities in the stratified sense. Further research could explore generalizations to more general types of singularities. Additionally, investigating the computational aspects of the derived formulas and their applications to specific examples would be of interest.

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Some L\^e-Greuel type formulae on stratified spaces

by Matthias Zac... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02682.pdf

Deeper Inquiries

How do these generalized Lé-Greuel formulas relate to other invariants of singularities in stratified spaces, such as the Goresky-MacPherson intersection homology Betti numbers?

The generalized Lé-Greuel formulas presented in the paper provide a powerful tool for computing the Milnor numbers µ(α;f), which capture the topology of stratified singularities. These numbers are intimately related to other invariants of stratified spaces, particularly the Goresky-MacPherson intersection homology Betti numbers.
Here's how the connection unfolds:

Intersection Homology and Stratified Morse Theory: Intersection homology is a homology theory specifically designed to study singular spaces. It refines the usual homology by considering only "allowable" chains that intersect the singular strata in a controlled manner. Stratified Morse theory, analogous to classical Morse theory, relates the topology of a stratified space to the critical points of a suitable function (a "Morse function"). The crucial point is that the Milnor fibers of the function at these critical points contribute to the intersection homology groups of the stratified space.

Milnor Numbers and Intersection Betti Numbers: The Milnor numbers µ(α;f) essentially count the number of spheres of various dimensions appearing in the bouquet decomposition of the Milnor fiber Mf∣(X,0). These spheres, in turn, contribute to the torsion-free part of the intersection homology groups of the stratified space. Therefore, the generalized Lé-Greuel formulas, by providing a way to compute the µ(α;f), indirectly give information about the intersection homology Betti numbers.

Beyond Torsion-Free Information: While the direct connection is with the torsion-free part of intersection homology, the Milnor numbers and the generalized Lé-Greuel formulas potentially encode more subtle information about the topology of the singularity. This is because the formulas involve complexes like the Koszul and Eagon-Northcott complexes, which are sensitive to the algebraic structure of the singularity.
In summary, the generalized Lé-Greuel formulas provide a bridge between the algebraic geometry of the singularity (encoded in the ideals and modules) and the topological invariants of the stratified space, including the intersection homology Betti numbers.

Could there be alternative expressions for the Milnor numbers µ(α;f) that do not rely on the Nash modification and instead use other resolutions of singularities?

It's certainly plausible that alternative expressions for the Milnor numbers µ(α;f) could be derived using resolutions of singularities other than the Nash modification. The key idea is that resolutions provide a way to "unfold" the singularity, making it easier to analyze its topology.
Here are some possibilities:

Blow-up Resolutions: Blowing up along subvarieties is a standard technique in algebraic geometry to resolve singularities. It's conceivable that by carefully choosing centers of blow-ups, one could obtain a resolution of the pair (X, f−1(0)) that is well-suited for computing the Milnor numbers. The challenge would be to relate the geometry of the exceptional divisors in the blow-up to the topology of the Milnor fiber.

Log Resolutions: Log resolutions are resolutions of singularities that have particularly nice properties with respect to divisors. They are often used in Hodge theory and related areas. It's possible that a log resolution of (X, f−1(0)) could lead to a more direct or computationally simpler expression for the µ(α;f).

Other Resolutions: Depending on the specific structure of the stratified space X and the function f, other specialized resolutions might be more suitable. For instance, if X has toric structure, toric resolutions could be advantageous.
The main advantage of using the Nash modification in the paper is its close relationship with the tangent spaces of the strata. This makes it relatively straightforward to define the obstruction cocycle and relate it to the degeneracy locus of the 1-forms. However, exploring alternative resolutions could potentially reveal new geometric insights and lead to formulas that are more computationally tractable in certain cases.

What insights from this work could be applied to the study of singularities in real algebraic geometry or the topology of real analytic sets?

While the paper focuses on complex analytic spaces, several insights and techniques could potentially be adapted to study singularities in real algebraic geometry or the topology of real analytic sets.
Here are some key points:

Real Analogues of Milnor Fibrations: The concept of a Milnor fibration has a natural counterpart in the real setting. For a real analytic function germ f : (ℝn, 0) → (ℝk, 0), one can consider the fibers of f over nearby regular values in ℝk. These fibers, while not necessarily having the same nice topological properties as in the complex case, still carry valuable information about the singularity.

Stratifications and Real Morse Theory: Real analytic sets admit Whitney stratifications, and real Morse theory can be applied to study their topology. The generalized Lé-Greuel formulas suggest that it might be fruitful to investigate "real" versions of these formulas, potentially involving invariants like the Euler characteristic modulo 2 or other suitable real analogues.

Connections to Real Intersection Homology: Goresky-MacPherson intersection homology is also defined for real algebraic varieties. Adapting the ideas from the paper could lead to new methods for computing intersection homology Betti numbers of real singular spaces.

Challenges in the Real Setting: It's important to note that working in the real setting presents additional challenges. For instance, real analytic sets can have more complicated local topology than their complex counterparts. Additionally, the lack of a canonical orientation in the real case might require modifications to the obstruction-theoretic arguments.
Despite these challenges, the ideas presented in the paper provide a valuable starting point for exploring the topology of real singularities. By carefully adapting the techniques and developing suitable real analogues of the key concepts, it might be possible to obtain new insights into the geometry and topology of real algebraic and analytic sets.