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A Lê number addendum to the Morse Splitting Lemma

This paper investigates the relationship between the complex analytic Morse Splitting Lemma and generic Lê numbers, specifically analyzing functions with a 0-dimensional generic Lê number of at most 3 or a 1-dimensional critical locus to determine what these conditions reveal about the function's structure.

Original authors: David B. Massey

Published 2026-08-25

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: David B. Massey

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to understanding the shapes of spaces defined by equations, particularly where those shapes twist, fold, or collapse into sharp points. Imagine a smooth, flowing surface that suddenly pinches together at a single spot; this is a singularity. Mathematicians study these points to understand the fundamental structure of the space around them. To do this, they often use a tool called the Morse Splitting Lemma, which acts like a powerful lens. It allows them to separate a complicated, knotted problem into two simpler parts: a smooth, predictable section and a messy, complex core that holds all the difficult behavior. For decades, researchers have used this separation to understand how functions behave near these sharp points, but a specific question remained about the "generic" nature of these points—what happens when we look at them from the most standard, unobstructed angle possible.

David B. Massey, a mathematician, has now provided a precise answer to this question by connecting the Morse Splitting Lemma to a set of measurements known as Lˆe numbers. These numbers are not physical quantities like weight or length, but rather counts that describe the intricate topology of the space near a singularity. They tell us how many distinct "holes" or loops exist in the tiny neighborhood surrounding a critical point, and how these features change as we move through the space. The core of Massey's work is a new theorem that acts as a supplement to the existing Morse Splitting Lemma. He proves that if a function has a certain level of complexity at a critical point, it can always be broken down into a simple, smooth quadratic part and a remaining function that carries all the heavy lifting. Crucially, the Lˆe numbers of the original, complicated function are exactly the same as the Lˆe numbers of this remaining, simpler function. This means that to understand the deep topological secrets of a complex shape, one only needs to study the smaller, leftover piece after the smooth parts have been peeled away.

The paper goes further by applying this finding to specific, low-complexity scenarios. Massey investigates what happens when the most basic Lˆe number, which counts a specific type of intersection, is very small—specifically, when it is one, two, or three. He demonstrates that if this number is one, the singularity is actually quite simple: it is a standard, non-degenerate point where the surface curves away cleanly in all directions, much like the bottom of a bowl. This result settles a previous uncertainty in the field, proving that it is impossible for a singularity with a one-dimensional line of critical points to have this specific low number. When the number is two or three, the paper shows that the function can still be split, but the remaining core function must have a specific minimum level of complexity, and its shape is constrained in a way that forces the surrounding space to look like a single sphere in a higher dimension.

In cases where the critical points form a line rather than a single dot, the paper clarifies exactly what it means to look at the shape from a "generic" or standard angle. It turns out that for these line-like singularities, the measurements are stable and predictable only if the viewing angle avoids certain specific alignments. Massey provides a concrete example to illustrate this: a function that looks simple from one angle but reveals hidden complexity from another. From a non-standard angle, the measurements might suggest a certain level of simplicity, but from the correct, generic angle, the true nature of the singularity is revealed to be even simpler. This distinction is vital because it ensures that mathematicians are measuring the intrinsic properties of the shape itself, rather than artifacts caused by the way they happen to be looking at it.

Ultimately, this work does not just add a new formula to a list; it tightens the logical connection between how we separate complex shapes and how we count their features. By proving that the Lˆe numbers remain invariant when the smooth parts of a function are removed, Massey gives researchers a reliable method to reduce difficult problems to their essential cores. The findings are not merely suggestions or simulations; they are rigorous proofs that hold true for all complex analytic functions within the defined scope. This allows the mathematical community to confidently state that for a wide range of singularities, the behavior of the whole is entirely determined by the behavior of a much smaller, manageable piece. The result is a clearer, more unified understanding of how complex shapes behave at their most fragile points, turning a collection of isolated observations into a coherent rule.

Technical Summary: A Lê Number Addendum to the Morse Splitting Lemma

Problem Statement
The paper investigates the relationship between the complex analytic Morse Splitting Lemma and the generic Lê numbers of a complex analytic function $f: (U, 0) \to (\mathbb{C}, 0)$ , where $U \subset \mathbb{C}^{n+1}$ . While the Morse Splitting Lemma describes the local structure of a function near a critical point based on the rank of its Hessian matrix, this work seeks to determine how this structural decomposition affects the generic Lê numbers ( $\lambda^*_f$ ). Specifically, the author addresses how the Lê numbers behave when the critical locus $\Sigma_f$ is non-isolated and aims to characterize functions where the 0-dimensional generic Lê number ( $\lambda^0_f(0)$ ) is small (at most 3) or where the critical locus has dimension 1.

Methodology
The paper employs techniques from complex analytic geometry and sheaf theory, specifically utilizing the theory of vanishing cycles and perverse sheaves.

Coordinate Selection: The analysis relies on the concept of "numerically generic" coordinates. These are linear coordinate choices at a point $p \in \Sigma_f$ where the tuple of Lê numbers attains its minimal value in lexicographical ordering. The author clarifies that while generic coordinates exist, a single coordinate system cannot be numerically generic for all points in a neighborhood; the focus is strictly on the point of interest (the origin).
Topological Proof: Rather than providing an algebraic derivation, the core of the methodology is a topological proof of a key lemma. This proof utilizes the Sebastiani-Thom Isomorphism within the derived category. It relates the stalk cohomology of the iterated nearby and vanishing cycles of a function $f$ to those of a reduced function $g$ after a coordinate change.
Intersection Theory: In analyzing specific cases (where $\lambda^0_f(0) \leq 3$ ), the paper uses intersection theory involving the relative polar curve $\Gamma^1_f$ and the hypersurface defined by the partial derivative $\partial f / \partial z_0$ .

Key Contributions and Results

The Main Theorem (Theorem 1.1): The paper establishes a "Lê number addendum" to the Morse Splitting Lemma. It proves that if the rank of the Hessian matrix of $f$ at the origin is $m \leq n$ , there exists an analytic change of coordinates $\eta$ such that $f$ decomposes as:
$(f \circ \eta)(w_1, \dots, w_m, u_1, \dots, u_{n-m}) = w_1^2 + \dots + w_m^2 + g(u_0, \dots, u_{n-m})$
Crucially, the generic Lê numbers of $f$ are identical to the generic Lê numbers of the residual function $g$ . Furthermore, the critical locus satisfies $\Sigma(f \circ \eta) = \{0\} \times \Sigma_g$ , and the multiplicity of $g$ at the origin is at least 3.
Proof of the Splitting Lemma for Lê Numbers (Lemma 1.2): The author proves that for a function of the form $f(w, y) = w^2 + g(y)$ , the critical locus is $\{0\} \times \Sigma_g$ and the generic Lê numbers of $f$ and $g$ at the origin are equal. This is demonstrated via the Sebastiani-Thom Isomorphism, showing the isomorphism of the relevant perverse sheaves restricted to their critical loci.
Classification for Small $\lambda^0_f(0)$ (Section 3):
- Case $\lambda^0_f(0) = 0$ : This implies the Milnor fiber is a product of a complex disk with the Milnor fiber of a slice, or the relative polar curve is empty.
- Case $\lambda^0_f(0) = 1$ : The author proves that if $\lambda^0_f(0) = 1$ , the critical locus must be 0-dimensional ( $s=0$ ), and $f$ has a non-degenerate isolated critical point with Milnor number 1. This resolves a previous uncertainty regarding whether $\lambda^0_f(0)$ could be 1 if the critical locus had dimension 1.
- Case $\lambda^0_f(0) = 2$ or $3$: The paper provides a structural decomposition for these cases, determining the possible multiplicities of the residual function $g$ and the homotopy type of the complex link of the singularity.
1-Dimensional Critical Loci (Section 4):
- The paper characterizes when Lê numbers are defined for 1-dimensional critical loci (requiring the coordinate $z_0$ to be "prepolar").
- It establishes that for $\dim \Sigma_f = 1$ , the 1-dimensional Lê cycle is independent of the coordinates, while the 0-dimensional Lê number depends on the coordinate choice.
- Proposition 4.1 provides necessary and sufficient conditions for coordinates to be numerically generic in the 1-dimensional case.
- Example 4.2 illustrates that coordinates can be prepolar (defining Lê numbers) without being numerically generic, showing a case where $\lambda^0_{f,u}(0) = 1$ (non-generic) versus $\lambda^0_{f,v}(0) = 0$ (generic).

Significance and Claims
The paper positions itself as a modest but precise supplement to existing literature. The author explicitly states that the primary result is a "Lê number supplement to the Morse Splitting Lemma." The significance lies in:

Bridging Structure and Invariants: It rigorously connects the algebraic decomposition of singularities (Morse Splitting) with topological invariants (Lê numbers), proving that the splitting preserves these invariants in the residual function.
Clarifying Genericity: It clarifies the distinction between the existence of generic coordinates and the specific "numerically generic" values required for calculation, particularly in the context of 1-dimensional critical loci.
Resolving Specific Cases: It definitively rules out the possibility of a 1-dimensional critical locus having a generic $\lambda^0_f(0) = 1$ , correcting a previous open question in the field.

The work does not propose new applications or future experimental directions but rather solidifies the theoretical framework for calculating and interpreting Lê numbers in the context of non-isolated singularities.

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