Higher characteristic classes of multisingularity loci

Abstract

A map between manifolds induces stratifications of both the source and the target according to the occurring multisingularities. In this paper, we study universal expressions-called higher Thom polynomials-that describe the Segre-Schwartz-MacPherson class of such multisingularity loci. We prove a Structure Theorem reducing these Thom polynomials to the data of a linear series associated with each multisingularity. The series corresponding to the empty multisingularity, referred to as the Master Series, plays a distinguished role. Motivated by connections with geometric representation theory, we further prove an Interpolation Theorem that allows Thom polynomials to be computed algorithmically within Mather's range of nice dimensions. As an application, we derive an explicit formula for the image Milnor number of quasihomogeneous germs, providing one side of the celebrated Mond conjecture, computable up to the theoretical bound.