Functions of Several Complex Variables and their Singularities
This book belongs to the AMS series ‘Graduate Studies in Mathematics’. Its main topic is a study of isolated singularities of hypersurfaces in several complex variables. The first chapter describes the classical theory of Riemann surfaces (coverings, fundamental groups, branched meromorphic continuation, the Riemann surfaces of an algebraic function and the Puiseux expansion). Basic facts of the theory of functions of several complex variables are explained in chapter 2 (the implicit function theorem, the Weierstrass preparation theorem and its generalizations, germs of analytic sets and their dimensions and special cases of the Remmert mapping theorem). Chapter 3 includes basic facts from differential geometry (smooth manifolds and their tangent bundles, transversality, homogeneous spaces and complex manifolds) together with a study of isolated critical points of holomorphic functions (the complex Morse lemma, universal unfolding, morsification and the classification of simple singularities).
Chapter 4 contains a description of basic facts from differential topology (the Ehresmann fibration theorem, a holonomy group of a fiber bundle, singular homology groups, the Euler characteristic, intersection and linking numbers, the braid group and the homotopy sequence of a fiber bundle). The last chapter is devoted to a description (partly without proofs) of topological properties of isolated critical points of holomorphic functions (including a discussion of monodromy groups and vanishing cycles, the Picard-Lefschetz theorem, the Milnor fibration, the intersection matrix and the Coxeter-Dynkin diagrams, variation of singularities, action of the braid group and the Arnold classification of unimodal singularities). The book contains a lot of illustrative pictures and diagrams substantially helping the geometric intuition of the reader.