Higher order double point formulas via SSM-Thom polynomials

TL;DR

This paper extends classical double point formulas for complex manifolds by incorporating Segre-Schwartz-MacPherson classes, providing a cohomological deformation that offers refined geometric insights and universal corrections.

Contribution

It introduces a one-parameter cohomological deformation of double point formulas using SSM classes and computes these classes for multisingularities, enhancing classical results with explicit universal corrections.

Findings

Recovered classical double point formula as leading term

Computed SSM classes for A0 and A1 singularities

Provided constraints on singularity loci as complete intersections

Abstract

We study the geometry of double point loci of maps $F:M\to N$ of complex manifolds through the lens of Segre-Schwartz-MacPherson (SSM) classes. Classical double point formulas express the fundamental class of the closure of the double point locus of $F$ in terms of global invariants of source and target spaces, as well as $F$. In this paper we extend these results by computing a one-parameter cohomological deformation of the double point formula given by the SSM class. We compute the SSM class of the double point locus in a large cohomological degree range. The leading term in our new formulas recovers the classical double point formula of Fulton and Laksov, while higher-degree terms provide explicit universal corrections. Our approach uses interpolation techniques for SSM-Thom polynomials of multisingularities, recently developed by Koncki, Nekarda, Ohmoto and Rim\'anyi. We also compute SSM-Thom polynomials for the singularities $A_0$ and $A_1$ in the same range. As an application, we show how the deformed formulas yield refined geometric information about those singularity loci through a theorem of Aluffi and Ohmoto, including constraints on when such loci can arise as complete intersections.