Introduction to Complex Analytic Geometry

facts. An elementary acquaintance with topology, algebra, and analysis (in­ cluding the notion of a manifold) is sufficient as far as the understanding of this book is concerned. All the necessary properties and theorems have been gathered in the preliminary chapters -either with proofs or with references to standard and elementary textbooks. The first chapter of the book is devoted to a study of the rings Oa of holomorphic functions. The notions of analytic sets and germs are introduced in the second chapter. Its aim is to present elementary properties of these objects, also in connection with ideals of the rings Oa. The case of principal germs (§5) and one-dimensional germs (Puiseux theorem, §6) are treated separately. The main step towards understanding of the local structure of analytic sets is Ruckert's descriptive lemma proved in Chapter III. Among its conse­ quences is the important Hilbert Nullstellensatz (§4). In the fourth chapter, a study of local structure (normal triples, § 1) is followed by an exposition of the basic properties of analytic sets. The latter includes theorems on the set of singular points, irreducibility, and decom­ position into irreducible branches (§2). The role played by the ring 0 A of an analytic germ is shown (§4). Then, the Remmert-Stein theorem on re­ movable singularities is proved (§6). The last part of the chapter deals with analytically constructible sets (§7). Preface to the Polish Edition Preface to the English Edition Contents 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. Rückert'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 Sadullaev theorems §8. Constructible sets. The Chevalley theorem §9. Rückert'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. Bézout's theorem. Biholomorphic mappings of projective spaces §14. Meromorphic functions and rational functions §15. Ideals of On with polynomial generators §l6. Serre's algebraic graph theorem. Zariski's analytic normality 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 [10a] [21] [35] NOTATION INDEX SUBJECT INDEX ab c de fgh i jklm nop qr st uvwz 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 C). Except for the last chapter, mostly local problems are investigated and, throughout the book, only the complex case is studied. From the purely geometric point of view, the real case is more natural and more general. But it displays fewer regularities and - by and large - the corresponding theory is more difficult. The complex structure is richer. Hence one can expect deeper results. Indeed, some phenomena, such as analyticity of the set of singular points (see IV. 24) or analyticity of proper images (Remmert's theorem, see V. 5.1), do not have counterparts in the real case. More than anything else, the beauty of the interplay between the geometric and algebraic phenomena constitutes the main attraction of the "complex" theory.This book should be regarded as an introduction. It does not pretend to reflect the entire theory. Its aim is to familiarize the reader with the basic range of problems, using means as elementary as possible. They belong tocomplex analysis, commutative algebra, and set topology (the methods of algebraic topology have not been employed), and are gathered in the first three preliminary chapters. 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 accesss to complete proofs without the 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 elementary textbooks.