Polynomials and vanishing cycles / Mihai Tibar

The book consists of preface with a short description of the content, three main parts, two appendices, a section of notes involving remarks and historical comments, two lists of references and bibliography containing 219 and 16 items, respectively, and the subject index.

The first part is devoted to the theory of polynomials with singularities at infinity. The author describes basic notions and results subsequently: atypical values, the control function and regularity, the Malgrange condition, polar curves, local and global fibrations, vanishing cycles and a method for computation of their number, basic properties of families of complex polynomials, the variation of the monodromy in such families, the contact structure at infinity, and others.

The second part deals with global polar varieties. In particular, the author describes in detail the Lefschetz slicing principle, the CW-complex structure, relative homology groups, the properties of global Euler obstruction, the concept of asymptotic equisingularity and topological triviality of families of affine hypersurfaces, a variant of Plücker’s formula for hypersurfaces with isolated singularities, the integral of the curvature and the Gauss-Bonnet defect for singular hypersurfaces, the relative monodromy and the zeta-function, etc.

The last part is concerned with topology of pencils of hypersurfaces on stratified complex spaces. In fact, such pencils can be considered as meromorphic functions. The author studies their singularities and monodromy, the vanishing homology, the slicing of the corresponding pencils and some applications concerning the Zariski-Lefschetz theorem and its generalizations. The first appendix contains a brief account of the theory of stratified singularities while the second one includes hints to exercises from the basic text. ... The most part of the book is written in a clear didactic style, with detailed explanations of all notions and results, almost all sections are followed by good exercises, comments and remarks. Therefore the book reads fluently, it is accessible for graduate students (especially its first two parts) while advanced researchers in topology, singularity theory and algebraic geometry can find here very interesting and useful material including some new results together with a new presentation of well-known concepts and proofs. (Zentralblatt)

Bibliogr. p. 247-248. Notes bibliogr. Index