Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chern-Schwartz-MacPherson classes in the point of view of Obstruction Theory and Lipschitz framework

Jean-Paul Brasselet, Tadeusz Mostowski, Thuy Nguyen Thi Bich

TL;DR

The paper reformulates Schwartz's obstruction-theoretic construction of Chern classes for singular complex varieties within Lipschitz stratifications, replacing Whitney stratifications to obtain a simpler, parallelizable framework. It introduces dual cellular decompositions and radial extensions to define obstruction cocycles, yielding Schwartz classes $c^p(X)$ in $H^{2p}({\\mathbb{C}}^m,{\\mathbb{C}}^m\setminus X)$ that are independent of stratification and triangulation. Through Alexander duality, these classes correspond to the MacPherson class of the constructible function ${\bf 1}_X$, linking the obstruction-theoretic view to MacPherson's theory and producing the Schwartz-MacPherson classes. The approach leverages Lipschitz stratifications to obtain continuity and coherence of $r$-frames and their radial extensions, simplifying the construction and broadening applicability to complex analytic varieties.

Abstract

Since Chern and Grothendieck, Chern's characteristic class theory has made significant progress. In particular with regard to the classes of singular varieties. Conjectured by Grothendieck and Deligne and demonstrated by MacPherson, Chern classes of singular varieties have been defined in several ways, such as using polar varieties, Lagrangian theory... However, the initial definition using obstruction theory, due to Marie-Hélène Schwartz, has been forgotten. Despite the simple ideas that enabled the obstruction definition, their implementation using Whitney stratifications requires delicate and technical constructions. In the present article, we show that in the Lipschitz framework, the ideas of Marie-Hélène Schwartz lead to a simplified definition and construction of Chern classes of complex analytic varieties.

Chern-Schwartz-MacPherson classes in the point of view of Obstruction Theory and Lipschitz framework

TL;DR

The paper reformulates Schwartz's obstruction-theoretic construction of Chern classes for singular complex varieties within Lipschitz stratifications, replacing Whitney stratifications to obtain a simpler, parallelizable framework. It introduces dual cellular decompositions and radial extensions to define obstruction cocycles, yielding Schwartz classes in that are independent of stratification and triangulation. Through Alexander duality, these classes correspond to the MacPherson class of the constructible function , linking the obstruction-theoretic view to MacPherson's theory and producing the Schwartz-MacPherson classes. The approach leverages Lipschitz stratifications to obtain continuity and coherence of -frames and their radial extensions, simplifying the construction and broadening applicability to complex analytic varieties.

Abstract

Since Chern and Grothendieck, Chern's characteristic class theory has made significant progress. In particular with regard to the classes of singular varieties. Conjectured by Grothendieck and Deligne and demonstrated by MacPherson, Chern classes of singular varieties have been defined in several ways, such as using polar varieties, Lagrangian theory... However, the initial definition using obstruction theory, due to Marie-Hélène Schwartz, has been forgotten. Despite the simple ideas that enabled the obstruction definition, their implementation using Whitney stratifications requires delicate and technical constructions. In the present article, we show that in the Lipschitz framework, the ideas of Marie-Hélène Schwartz lead to a simplified definition and construction of Chern classes of complex analytic varieties.

Paper Structure

This paper contains 15 sections, 8 theorems, 51 equations, 5 figures.

Table of Contents

  1. Introduction.
  2. A few on obstruction theory
  3. The smooth case
  4. The singular case
  5. Stratifications
  6. Triangulations and cellular decompositions
  7. Obstruction in a stratified space
  8. Lipschitz stratifications
  9. Lipschitz vector fields
  10. The parallel Lipschitz extension
  11. The basic normal Lipschitz vector field
  12. "Champs sortants" in Lipschitz stratifications
  13. Applications
  14. Radial extension of $r$-frames
  15. Obstruction cocycles and classes

Key Result

Lemma 1.2

(i) The dual cell of a $k$-simplex is a $(2m-k)$-cell, homeomorphic to the unit ball ${\mathbb B}^{2m-k} \subset {\mathbb R}^{2m-k}$ and its boundary is homeomorphic to the corresponding sphere ${\mathbb{S}}^{2m-k-1}$. (ii) Given a triangulation $(K)$ of ${\mathbb C}^m$ compatible with the stratific

Figures (5)

  • Figure 1: Dual cells: The barycenter of $\sigma_0 = \{A \}$ is $A$ itself. The dual cell of $\sigma_0$ is the dark gray cell. The dual cell of $\sigma_1=AB$ is composed of the two segments (double lines) $A' \widehat{\sigma_1}$ and $\widehat{\sigma_1} B'$. The dual cell of the triangle $\sigma_2= ABC$ is the barycenter of $\sigma_2$ = $A'$. The stratum $S^j$ is the thick dashed line
  • Figure 2: Example of chain: $j_1= 2, j_t= j_2 = 1, j_3=0$, the chain is: $(q_{j_1}, \;q_{j_2},\;q_{j_3})$.
  • Figure 3: The parallel Lipschitz extension.
  • Figure 4: The neighbourhood $U$.
  • Figure 5: The starting point of induction (real picture of a complex situation). The cell $d$ is dual of the simplex $\sigma = ABC$.

Theorems & Definitions (21)

  • Lemma 1.2
  • Lemma 1.3
  • Remark 1.4
  • Definition 2.1: Definition of chains
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Lipschitz stratification
  • Remark 2.6
  • Remark 2.7
  • ...and 11 more