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Higher characteristic classes of multisingularity loci

Jakub Koncki, Richárd Rimányi

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.

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.
Paper Structure (45 sections, 42 theorems, 155 equations, 3 figures)

This paper contains 45 sections, 42 theorems, 155 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Multisingularity loci
  3. Universal polynomials: the Thom principle
  4. Segre–Schwartz–MacPherson class
  5. SSM–Thom polynomials for multisingularities
  6. The Master Series KazT(varnothing)
  7. Computations
  8. Mond conjecture
  9. Conventions
  10. Acknowledgments
  11. Singularities
  12. Contact monosingularities
  13. Multisingularities
  14. Multisingularity loci
  15. Prototypes of singularities, multisingularities
  16. ...and 30 more sections

Key Result

Proposition 4.1

Let $M$ and $N$ be smooth varieties.

Figures (3)

  • Figure 1: The interrelations among the various concepts of Section \ref{['sec:induction']}.
  • Figure 2: $S_{\underline{\eta}}$ polynomials for $l=1$ up to cohomological degree 6
  • Figure 3: $R_{\underline{\eta}}$ polynomials for $l=1$ up to cohomological degree 5. Recall that $R_{{\underline{\eta}}}$ does not depend on the distinguished element of ${\underline{\eta}}$.

Theorems & Definitions (111)

  • Remark 1.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6: Multisingularity induced by subset
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • ...and 101 more