CiNii Books - Polynomials and vanishing cycles

The behaviour of vanishing cycles is the cornerstone for understanding the geometry and topology of families of hypersurfaces, usually regarded as singular fibrations. This self-contained tract proposes a systematic geometro-topological approach to vanishing cycles, especially those appearing in non-proper fibrations, such as the fibration defined by a polynomial function. Topics which have been the object of active research over the past 15 years, such as holomorphic germs, polynomial functions, and Lefschetz pencils on quasi-projective spaces, are here shown in a new light: conceived as aspects of a single theory with vanishing cycles at its core. Throughout the book the author presents the current state of the art. Transparent proofs are provided so that non-specialists can use this book as an introduction, but all researchers and graduate students working in differential and algebraic topology, algebraic geometry, and singularity theory will find this book of great use.

Table of Contents

Preface
Part I. Singularities at Infinity of Polynomial Functions: 1. Regularity conditions at infinity
2. Detecting atypical values via singularities at infinity
3. Local and global fibrations
4. Families of complex polynomials
5. Topology of a family and contact structures
Part II. The Impact of Global Polar Varieties: 6. Polar invariants and topology of affine varieties
7. Relative polar curves and families of affine hypersurfaces
8. Monodromy of polynomials
Part III. Vanishing Cycles of Non-Generic Pencils: 9. Topology of meromorphic functions
10. Slicing by pencils of hypersurfaces
11. Higher Zariski-Lefschetz theorems
Notes
References
Bibliography
Appendix 1. Stratified singularities
Appendix 2. Hints to exercises
Index.

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Polynomials and vanishing cycles

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