# Numerical Control over Complex Analytic Singularities

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*David B. Massey*

The Milnor number is a powerful invariant of an isolated,
complex, affine hypersurface singularity. It provides data about the local,
ambient, topological-type of the hypersurface, and the constancy of the Milnor
number throughout a family implies that Thom's \(a_f\) condition holds
and that the local, ambient, topological-type is constant in the family. Much
of the usefulness of the Milnor number is due to the fact that it can be
effectively calculated in an algebraic manner.

The Lê cycles and numbers are a generalization of the
Milnor number to the setting of complex, affine hypersurface singularities,
where the singular set is allowed to be of arbitrary dimension. As with the
Milnor number, the Lê numbers provide data about the local, ambient,
topological-type of the hypersurface, and the constancy of the Lê numbers
throughout a family implies that Thom's \(a_f\) condition holds and that
the Milnor fibrations are constant throughout the family. Again, much of the
usefulness of the Lê numbers is due to the fact that they can be
effectively calculated in an algebraic manner.

In this work, we generalize the Lê cycles and numbers to
the case of hypersurfaces inside arbitrary analytic spaces. We define the
Lê-Vogel cycles and numbers, and prove that the Lê-Vogel numbers
control Thom's \(a_f\) condition. We also prove a relationship between
the Euler characteristic of the Milnor fibre and the Lê-Vogel numbers.
Moreover, we give examples which show that the Lê-Vogel numbers are
effectively calculable.

In order to define the Lê-Vogel cycles and numbers, we
require, and include, a great deal of background material on Vogel cycles,
analytic intersection theory, and the derived category. Also, to serve as a
model case for the Lê-Vogel cycles, we recall our earlier work on the
Lê cycles of an affine hypersurface singularity.

#### Table of Contents

# Table of Contents

## Numerical Control over Complex Analytic Singularities

- Table of Contents vii8 free
- Abstract ix10 free
- Preface xi12 free
- Overview 114 free
- Part I. Algebraic Preliminaries: Gap Sheaves and Vogel Cycles 316
- Part II. Lê Cycles and Hypersurface Singularities 3750
- Introduction 3750
- Chapter 1. Definitions and Basic Properties 4457
- Chapter 2. Elementary Examples 6578
- Chapter 3. A Handle Decomposition of the Milnor Fibre 7184
- Chapter 4. Generalized Lê-Iomdine Formulas 7588
- Chapter 5. Le Numbers and Hyperplane Arrangements 91104
- Chapter 6. Thom's a[sub(f)] Condition 97110
- Chapter 7. Aligned Singularities 103116
- Chapter 8. Suspending Singularities 109122
- Chapter 9. Constancy of the Milnor Fibrations 114127
- Chapter 10. Another Characterization of the Lê Cycles 120133

- Part III. Isolated Critical Points of Functions on Singular Spaces 123136
- Chapter 0. Introduction 123136
- Chapter 1. Critical Avatars 127140
- Chapter 2. The Relative Polar Curve 133146
- Chapter 3. The Link between the Algebraic and Topological Points of View 139152
- Chapter 4. The Special Case of Perverse Sheaves 146159
- Chapter 5. Thom's a[sub(f)]Condition 154167
- Chapter 6. Continuous Families of Constructible Complexes 160173

- Part IV. Non-Isolated Critical Points of Functions on Singular Spaces 175188
- Appendix A. Analytic Cycles and Intersections 195208
- Appendix B. The Derived Category and Vanishing Cycles 203216
- Appendix C. Privileged Neighborhoods and Lifting Milnor Fibrations 243256
- References 261274
- Index 267280