Thom polynomials for singularities of maps

TL;DR

This article surveys Thom polynomial theory as a universal framework for counting and obstructing singularities of map-germs. It foregrounds equivariant cohomology and localization, along with desingularization and Hilbert-scheme approaches, to construct and compute universal polynomials $tp(\eta)$ (and their higher/SM variants) that express singularity loci in terms of Chern data. It covers single-singularity classifications ($\mathcal{K}$-, $\mathcal{A}$-, Lagrange/Legendre) and extends to multi-singularities via Kazarian’s residual-polynomial program, with substantial developments in Thom series, Schubert-calculus, and cobordism frameworks. The survey also highlights important applications in enumerative geometry, vanishing topology, and complex/hyperbolic geometry, including recent polynomial Green-Griffiths-Lang results obtained via non-reductive GIT. Overall, Thom polynomials provide a unifying, computational toolkit connecting local singularity types to global geometric invariants across complex and real settings.

Abstract

This is a gentle introduction to a general theory of universal polynomials associated to classification of map-germs, called Thom polynomials. The theory was originated by René Thom in the 1950s and has since been evolved in various aspects by many authors. In a nutshell, this is about intersection theory on certain moduli spaces, say `classifying spaces of mono/multi-singularities of maps', which provides consistent and deep insights into both classical and modern enumerative geometry with many potential applications.

Figures (1)

• Figure 1: Singularities of plane-to-plane maps: fold, cusp and double-folds