Polynomials and Vanishing Cycles
This monograph is devoted to the study of families of hypersurfaces considered as singular fibrations. It is based on the author’s own research over the last 12 years. Let us cite here the author’s words from the introduction: “The leading idea of this monograph is to bring into new light a bunch of topics – holomorphic germs, polynomial functions, pencils on quasi-projective spaces - conceiving them as aspects of a single theory with vanishing cycles at its core”. Really, the notions of vanishing homology and vanishing cycles play a very important role here. The book is divided into three parts. Parts I and II deal with complex polynomial functions and in the very centre of the theory, we find vanishing cycles at infinity. Counting of vanishing cycles is closely related to polar curves, which are investigated in part II. Here we come to important invariants of affine varieties such as CW-complex structure, relative homology, Euler obstruction and the Chern-MacPherson cycle. In part III the author passes to the meromorphic situation and studies the topology of pencils of hypersurfaces. The book is intended for specialists in the field and for graduate students. The exposition is rather self-contained. Nevertheless it requires some preliminary knowledge including some differential and algebraic topology, some algebraic geometry and some familiarity with Milnor’s classical book “Singular points of complex hypersurfaces”. Each chapter of the book is followed by exercises and sometimes there are hints on how to solve them.