Polynomials and Vanishing Cycles
Polynomials and Vanishing Cycles
$142.00
USDThe 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.
- Self-contained and accessible tract proposing a systematic geometro-topological approach to vanishing cycles appearing in non-proper fibrations
- The latest research on new topics such as topology of singularities of meromorphic functions and non-generic Lefschetz pencils is discussed in detail
- Presents the current state of the art in this field
Reviews & endorsements
'… written in a clear didactic style, with detailed explanations of all notions and results … the book reads fluently, it is accessible for graduate students while advanced researchers … can find here very interesting and useful material …' Zentralblatt MATH
'Each chapter of the book is followed by exercises …' EMS Newsletter
Product details
June 2007Hardback
9780521829205
266 pages
235 × 160 × 25 mm
0.568kg
12 b/w illus. 3 tables 53 exercises
Available
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.
