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Numerical Control over Complex Analytic Singularities
David B. Massey, Northeastern University, Boston, MA
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Memoirs of the American Mathematical Society
2003; 268 pp; softcover
Volume: 163
ISBN-10: 0-8218-3280-8
ISBN-13: 978-0-8218-3280-6
List Price: US$84 Individual Members: US$50.40
Institutional Members: US\$67.20
Order Code: MEMO/163/778

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.

Graduate students and research mathematicians interested in several complex variables and analytic spaces.

• Overview
Part I. Algebraic Preliminaries: Gap Sheaves and Vogel Cycles
• Introduction
• Gap sheaves
• Gap cycles and Vogel cycles
• The Lê-Iomdine-Vogel formulas
• Summary of Part I
Part II. Lê Cycles and Hypersurface Singularities
• Introduction
• Definitions and basic properties
• Elementary examples
• A handle decomposition of the Milnor fibre
• Generalized Lê-Iomdine formulas
• Lê numbers and hyperplane arrangements
• Thom's $$a_f$$ condition
• Aligned singularities
• Suspending singularities
• Constancy of the Milnor fibrations
• Another characterization of the Lê cycles
Part III. Isolated Critical Points of Functions on Singular Spaces
• Introduction
• Critical avatars
• The relative polar curve
• The link between the algebraic and topological points of view
• The special case of perverse sheaves
• Thom's $$a_f$$ condition
• Continuous families of constructible complexes
Part IV. Non-Isolated Critical Points of Functions on Singular Spaces
• Introduction
• Lê-Vogel cycles
• Lê-Iomdine formulas and Thom's condition
• Lê-Vogel cycles and the Euler characteristic
• Appendix A. Analytic cycles and intersections
• Appendix B. The derived category
• Appendix C. Privileged neighborhoods and lifting Milnor fibrations
• References
• Index