# Introduction to Complex Analytic Geometry

U.S. Government Printing Office, 1991 - Mathematics - 523 pages
The subject of this book is analytic geometry, understood as the geometry of analytic sets (or, more generally, analytic spaces), i.e. sets described locally by systems of analytic equations. Though many of the results presented are relatively modern, they are already part of the classical tool-kit of workers in analytic and algebraic geometry and in analysis, for example: the theorems of Chevalley on constructible sets, of Remmert-Stein on removable singularities, of Andreotti-Stoll on the fibres of a finite mapping, and of Andreotti-Salmon on factoriality of the Grassmannian. The chapter on the relationship between analytic and algebraic geometry is particularly illuminating. This book should be regarded as an introduction. Its aim is to familiarize the reader with a basic range of problems, using means as elementary as possible. At the same time, the author's intention is to give the reader access to complete proofs without need to rely on so-called 'well- known' facts. All the necessary properties and theorems have been gathered in the first chapters - either with proofs or with references to standard and elemenatry textbooks. Contents Preface to the Polish Edition Preface to the English Edition PRELIMINARIES Chapter A. Algebra -1. Rings, fields, modules, ideals, vector spaces -2. Polynomials -3. Polynomial mappings -4. Symmetric polynomials. Discriminant -5. Extensions of fields -6. Factorial rings -7. Primitive element theorem -8. Extensions of rings -9. Noetherian rings -10. Local rings -11. Localization -12. Krull's dimension -13. Modules of syzygies and homological dimension -14. The depth of a module -15. Regular rings Chapter B. Topology -1. Some topological properties of sets and families of sets -2. Open, closed and proper mappings -3. Local homeomorphisms and coverings -4. Germs of sets and functions -5. The topology of a finite dimensional vector space (over C or R ) -6. The topology of the Grassmann space Chapter C. Complex analysis -1. Holomorphic mappings -2. The Weierstrass preparation theorem -3. Complex manifolds -4. The rank theorem. Submersions COMPLEX ANALYTIC GEOMETRY Chapter I. Rings of germs of holomorphic functions -1. Elementary properties. Noether and local properties. Regularity -2. Unique factorization property -3. The Preparation Theorem in Thom-Martinet version Chapter II. Analytic sets, analytic germs and their ideals -1. Dimension -2. Thin sets -3. Analytic sets and germs -4. Ideals of germs and the loci of ideals. Decomposition into simple germs -5. Principal germs -6. One-dimensional germs. The Puiseux theorem Chapter III. Fundamental lemmas -1. Lemmas on quasi-covers -2. Regular and K-normal ideals and germs -3. Rueckert's descriptive lemma -4. Hilbert's Nullstellensatz and other consequences(concerning dimension, regularity and K-normality) Chapter IV. Geometry of analytic sets -1. Normal triples -2. Regular and singular points. Decomposition into simple components -3. Some properties of analytic germs and sets -4. The ring of an analytic germ. Zariski's dimension -5. The maximum principle -6. The Remmert-Stein removable singularity theorem -7. Regular separation -8. Analytically constructible sets Chapter V. Holomorphic mappings -1. Some properties of holomorphic mappings of manifolds -2. The multiplicity theorem. Rouche's theorem -3.Holomorphic mappings of analytic sets -4. Analytic spaces -5. Remmert's proper mapping theorem -6. Remmert's open mapping theorem -7. Finite holomorphic mappings -8. c-holomorphic mappings Chapter VI. Normalization -1. The Cartan and Oka coherence theorems -2. Normal spaces. Universal denominators -3. Normal points of analytic spaces -4. Normalization Chapter VII. Analyticity and algebraicity -1. Algebraic sets and their ideals -2. The projective space as a manifold -3. The projective closure of a vector space -4. Grassmann manifolds -5. Blowings-up -6. Algebraic sets in projective spaces. Chow's theorem -7. The Rudin and Sudallaev theorems -8. Constructible sets. The Chevalley theorem -9. Rueckert's lemma for algebraic sets -10. Hilbert's Nullstellensatz for polynomials -11. Further properties of algebraic sets. Principal varieties. Degree -12. The ring of an algebraic subset of a vector space -13. Bezout's theorem. Biholomorphic mappings of projective spaces -14. Meromorphic functions and rational functions -15. Ideals of On with polynomial generators -16. Serre's algebraic graph theorem -17. Algebraic spaces -18. Biholomorphic mappings of factorial subsets in projective spaces -19. The Andreotti-Salmon theorem -20. Chow's theorem on biholomorphic mappings of Grassmann manifolds References Notation index Subject index

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### Contents

 PRELIMINARIES CHAPTER A Algebra 1 Rings fields modules ideals vector spaces 1 Polynomials 3 Polynomial mappings 4 Symmetric polynomials Discriminant 25