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Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularities

Thaís M. Dalbelo, Hellen Santana

TL;DR

This work extends Morse-theoretic ideas to non-degenerate locally tame complete intersections by introducing Newton-admissible families and induced good stratifications, enabling a Lê-Greuel-type relation between Brasselet numbers and Morse critical points on the Milnor fiber even when the critical locus has arbitrary dimension. It provides explicit, polyhedral formulas for Brasselet numbers in terms of volumes of Newton polyhedra, leveraging Matsui–Takeuchi results, and proves invariance of Morse counts and Brasselet numbers under small Newton-admissible perturbations. The approach unifies topological invariants with combinatorial Newton-data in toric and non-toric complete intersections, yielding practical criteria for equisingularity and computable invariants via polyhedral geometry. The results generalize prior isolated- and one-dimensional cases and open pathways for algorithmic computation of invariants in singularity theory.

Abstract

The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geo\-me\-tri\-cal study of singular spaces. Let \{$X_t\}_t$ be a suitable family of germs at $0$ of complete intersection varieties in $\mathbb{C}^n$ and $\{f_t\}_t, \{g_t\}_t$ families of non-constant polynomial functions on $X_t$. If the germs $X_t$, $X_t \cap f_t^{-1}(0)$ and $X_t\cap f_t^{-1}(0) \cap g_t^{-1}(0)$ are non-degenerate, locally tame, complete intersection varieties, for each $t,$ we prove that the difference of the Brasselet numbers, ${\rm B}_{f_t,X_t}(0)$ and ${\rm B}_{f_t,X_t\cap g_t^{-1}(0)}(0)$, is related with the number of Morse critical points {on the regular part of the Milnor fiber} of $f_t$ appearing in a morsefication of $g_t$, even in the case where $g_t$ has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.

Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularities

TL;DR

This work extends Morse-theoretic ideas to non-degenerate locally tame complete intersections by introducing Newton-admissible families and induced good stratifications, enabling a Lê-Greuel-type relation between Brasselet numbers and Morse critical points on the Milnor fiber even when the critical locus has arbitrary dimension. It provides explicit, polyhedral formulas for Brasselet numbers in terms of volumes of Newton polyhedra, leveraging Matsui–Takeuchi results, and proves invariance of Morse counts and Brasselet numbers under small Newton-admissible perturbations. The approach unifies topological invariants with combinatorial Newton-data in toric and non-toric complete intersections, yielding practical criteria for equisingularity and computable invariants via polyhedral geometry. The results generalize prior isolated- and one-dimensional cases and open pathways for algorithmic computation of invariants in singularity theory.

Abstract

The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geo\-me\-tri\-cal study of singular spaces. Let \{ be a suitable family of germs at of complete intersection varieties in and families of non-constant polynomial functions on . If the germs , and are non-degenerate, locally tame, complete intersection varieties, for each we prove that the difference of the Brasselet numbers, and , is related with the number of Morse critical points {on the regular part of the Milnor fiber} of appearing in a morsefication of , even in the case where has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.

Paper Structure

This paper contains 7 sections, 15 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions and results
  3. Euler obstruction
  4. Stratifications and Brasselet number
  5. Non-degenerate locally tame complete intersection variety
  6. Induced good stratifications for non-degenerate locally tame complete intersections
  7. Euler obstruction, Morse points and torus action

Key Result

Theorem 2.2

Let $(X,0)$ and $\mathcal{V}$ be given as before, then for each generic linear form $l$, there exists $\varepsilon_0$ such that for any $\varepsilon$ with $0<\varepsilon<\varepsilon_0$ and $\delta\neq0$ sufficiently small, the Euler obstruction of $(X,0)$ is equal to where $\chi$ is the Euler characteristic, ${\rm Eu}_{X}(V_i)$ is the Euler obstruction of $X$ at a point of $V_i, \ i=1,\ldots,q$ a

Theorems & Definitions (34)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4: BMPS, Theorem 3.1
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 24 more