Universal polynomials for multi-singularity loci of maps

TL;DR

This paper proves the existence of universal polynomials, called Thom polynomials, that describe multi-singularity loci classes for proper morphisms between smooth schemes, advancing the enumerative theory of singularities in algebraic geometry.

Contribution

It establishes the Thom-Kazarian principle by proving the existence of Thom polynomials for multi-singularity types, providing a foundational tool for enumerative geometry of singularities.

Findings

Existence of universal Thom polynomials for multi-singularity loci.

Application of algebro-geometric cohomology operations in the proof.

Connection to algebraic cobordism and motivic cohomology for classical enumerative geometry.

Abstract

In the present paper, we prove the existence of universal polynomials which express multi-singularity loci classes of prescribed types for proper morphisms between smooth schemes over an algebraically closed field of characteristic zero -- we call them Thom polynomials for multi-singularity types of maps. It has been referred to as the Thom-Kazarian principle and unsolved for a long time. This result solidifies the foundation for a general enumerative theory of singularities of maps which is applicable to a broad range of problems in classical and modern algebraic geometry. In particular, it would contribute to a satisfactory answer to the rest of (an advanced form of) Hilbert's 15th problem and connect such classics to recent new interests in enumerations inspired by mathematical physics and other fields. A main feature of our proof is a striking use of algebro-geometric cohomology operations. Somewhat surprisingly, when trying to grasp a full perspective of classical enumerative geometry, we will inevitably encounter algebraic cobordism and motivic cohomology.