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Thom polynomials for singularities of maps

This paper provides a gentle introduction to the theory of Thom polynomials, which are universal polynomials used to study the classification of map-germs through intersection theory on specific moduli spaces.

Original authors: Toru Ohmoto

Published 2026-02-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Toru Ohmoto

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

Imagine you are a photographer trying to capture the perfect landscape. Sometimes, your lens is perfect, and the image is crisp. But sometimes, you encounter "glitches": a smudge on the lens, a blur because the subject moved, or a strange distortion where two different parts of the landscape seem to overlap in a way they shouldn't.

In mathematics, specifically in the study of mappings (functions that take points from one space and move them to another), these "glitches" are called singularities.

This paper, written by Toru Ohmoto, is a high-level map of a mathematical field called Thom Polynomial Theory. It explains how mathematicians can predict, count, and categorize these "glitches" using special formulas.

Here is the breakdown of the paper using everyday analogies.

1. The Core Idea: The "Glitch Predictor" (Thom Polynomials)

Imagine you are running a massive factory line that produces thousands of glass vases every hour. You want to know: "How many vases will have a crack in them if the machine vibrates at a certain frequency?"

Instead of waiting for the cracks to happen and counting them one by one, you want a formula. You want to plug in the "vibration level" (the properties of your map) and have the formula spit out the "number of cracks" (the number of singularities).

Thom Polynomials are those formulas. They are "universal" because they don't care about the specific shape of your vase; they only care about the type of crack you are looking for. Whether you are making a vase, a car, or a digital animation, if the "glitch" is the same type, the formula works the same way.

2. The Different Types of Glitches (Classification)

The paper explains that not all glitches are the same. It categorizes them into different "flavors":

The Fold (A1): Imagine folding a piece of paper. The edge where the paper bends is a "fold." It’s a simple, predictable line of singularity.
The Cusp (A2): Imagine a piece of paper being pinched into a sharp point. This is more complex than a fold; it’s a "cusp."
Multi-singularities: This is when glitches happen at the same time or in the same place. Imagine two different cracks in a vase meeting at a single point. This is much harder to predict, but the paper discusses how we can still use math to track these "double" or "triple" glitches.

3. The Tools: How do we solve the puzzle?

The paper describes several "mathematical toolkits" used to find these formulas:

The "Interpolation" Method (The Detective Work): Instead of building a whole new machine to find cracks, you look at a few known broken vases, study their patterns, and "interpolate" (fill in the blanks) to figure out the master formula.
The "Localization" Method (The Spotlight): Imagine a dark room full of moving parts. Instead of trying to see everything, you shine a very bright, specific spotlight on just one tiny spot. By seeing how the light behaves at that one point, you can deduce the rules for the whole room.
The "Hilbert Scheme" (The Catalog): This is like a massive, organized library that contains every possible way a "glitch" can exist. If you know which "book" your glitch belongs to, you can find its formula.

4. Why does this matter? (Applications)

This isn't just abstract "math for math's sake." The paper points out that these formulas help in:

Art and Design: Understanding how shapes distort when projected onto a screen (like in CGI or computer graphics).
Physics: Counting "instantons"—tiny, complex structures in the fabric of the universe.
Geometry: Solving ancient puzzles about how many points on a curve are "special" (like inflection points).

Summary: The Big Picture

If the universe is a giant, complex movie, singularities are the moments where the film glitches, blurs, or folds. Thom Polynomial Theory is the mathematical science of writing the "code" that predicts exactly when and how those glitches will appear, allowing us to understand the underlying structure of the movie itself.

This paper, "Thom Polynomials for Singularities of Maps" by Toru Ohmoto, serves as a comprehensive survey and state-of-the-art report on the theory of Thom polynomials. It traces the evolution of the field from René Thom’s foundational work in the 1950s to modern developments involving equivariant cohomology, Hilbert schemes, and non-reductive Geometric Invariant Theory (GIT).

1. The Problem: Enumerating Singularities

The central problem addressed is the enumeration of singularities of prescribed types in differentiable or holomorphic mappings between manifolds. When a map $f: M \to N$ is not a submersion, it possesses singular points (where the differential is degenerate) or multiple image points.

The goal is to find universal cohomological obstructions—polynomials in the characteristic classes of the tangent bundles of $M$ and $N$ —that represent the locus of these singular points. These "Thom polynomials" allow mathematicians to calculate the number (or the homology class) of singular points of a specific type (e.g., folds, cusps, or multi-singularities) based solely on the global topology of the source and target manifolds.

2. Methodology: From Desingularization to Localization

The paper outlines three major methodological shifts in the history of the theory:

The Classical Desingularization Method: Early approaches relied on explicitly constructing a "resolution of singularities" (e.g., via flag bundles or intrinsic derivatives) to compute the Gysin image of the top Chern class of a vector bundle. This method is geometrically intuitive but becomes computationally intractable for higher-order singularities.
The Interpolation (Restriction) Method: Developed significantly by R. Rimányi, this method uses equivariant cohomology. By considering the action of a Lie group $G$ on a jet space $V$ , the Thom polynomial is determined by its restriction to the $G$ -orbits. This transforms the problem into solving a system of linear equations based on the normal Euler classes of the orbits.
The Localization and Hilbert Scheme Method: The most modern approach (Bérczi-Szenes) utilizes torus-equivariant localization and the geometry of punctual Hilbert schemes. By viewing singularity types as strata in a Hilbert scheme, the Thom polynomial can be expressed as an iterated residue integral. This provides a powerful, algorithmic way to handle singularities with arbitrary codimension $\kappa$ .

3. Key Contributions and Results

A. Classification and Universality

The paper distinguishes between different equivalence relations:

K-classification (Contact equivalence): Leads to stable Thom polynomials that depend only on the relative codimension $\kappa$ and are expressed via quotient Chern classes.
A-classification (Right-left equivalence): More restrictive; the polynomials depend on both the source and target Chern classes separately.

B. Higher Thom Polynomials (SSM Classes)

Ohmoto highlights the extension of the theory to higher Thom polynomials using Chern-Schwartz-MacPherson (CSM) classes and Segre-Schwartz-MacPherson (SSM) classes. While standard Thom polynomials represent the fundamental cycle of a singularity locus, higher Thom polynomials capture deeper topological information (like the Euler characteristic or Milnor numbers) of singular varieties.

C. Multi-singularities

The paper discusses the Thom-Kazarian principle, which provides a framework for "multi-singularities" (where multiple points in the source map to the same point in the target). It introduces residual polynomials and shows how the source and target loci are related through complex combinatorial formulas.

D. Applications

The survey demonstrates the utility of the theory in:

Enumerative Geometry: Counting nodal or cuspidal curves in a linear system (instanton counting).
Vanishing Topology: Calculating Milnor numbers and image Milnor numbers of map-germs.
Complex Hyperbolic Geometry: A surprising connection is noted where the positivity of Thom polynomial coefficients (the Rimányi conjecture) relates to the existence of holomorphic curves on hypersurfaces (Green-Griffiths-Lang and Kobayashi conjectures).

4. Significance

The significance of this paper lies in its ability to unify disparate fields:

Singularity Theory & Topology: It provides the tools to link local algebraic properties of map-germs to the global topology of manifolds.
Algebraic Geometry & Representation Theory: It connects the study of degeneracy loci to Schubert calculus and the geometry of moduli spaces (Hilbert schemes).
Mathematical Physics: It provides the mathematical foundation for counting instantons and understanding the geometry of discriminants.

Conclusion: Ohmoto presents Thom polynomial theory not merely as a tool for counting, but as a deep, evolving intersection of equivariant geometry, commutative algebra, and complex analysis that continues to provide insights into the fundamental nature of mappings.