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Some Lê-Greuel type formulae on stratified spaces

Matthias Zach

Abstract

We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$. An extension of Tib{\u a}r's Bouquet Theorem to this setup allows for a topological definition of Milnor numbers $μ(α; f)$ for each stratum $V^α$ of $X$ and we prove several formulas which compute these numbers as (alternating) sums of certain ``homological indices''. The main technical result at work in the background is a local Riemann-Roch type theorem, relating a topological obstruction to holomorphic Euler characteristics.

Some Lê-Greuel type formulae on stratified spaces

Abstract

We extend the circle of ideas from a previous paper on hypersurfaces to functions with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ . An extension of Tib{\u a}r's Bouquet Theorem to this setup allows for a topological definition of Milnor numbers for each stratum of and we prove several formulas which compute these numbers as (alternating) sums of certain ``homological indices''. The main technical result at work in the background is a local Riemann-Roch type theorem, relating a topological obstruction to holomorphic Euler characteristics.

Paper Structure

This paper contains 19 sections, 18 theorems, 138 equations, 1 table.

Table of Contents

  1. Introduction and results
  2. The classical Lê-Greuel Formula
  3. The central new results
  4. Generic complexes and homological indices
  5. Topological preliminaries
  6. A brief review of the Handlebody Theorem
  7. Morsifications and homological decompositions
  8. Generalization to complete intersections
  9. Obstructions on fibers of maps
  10. Collections of $1$-forms on stratified spaces
  11. Obstruction theory
  12. Nash modifications
  13. The main technical result
  14. Geometric interpretations of some commutative algebra
  15. The Koszul complex
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $(X, 0) \subset (\mathbb{C}^n, 0)$ be a reduced complex analytic space of dimension $\dim (X, 0) \geq 2$It is not explicitly stated but seemingly understood from the proof in Tibar95 that actually no component of $(X, 0)$ must have dimension $<2$, endowed with a Whitney stratification $X = \bigc is a complex analytic germ such that the restriction $f|_X$ has an isolated singularity in the stra

Theorems & Definitions (43)

  • Theorem 2.1
  • Example 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Example 2.5
  • Theorem 2.6
  • proof
  • Definition 2.7
  • Theorem 2.8
  • Example 2.9
  • ...and 33 more