Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex Analysis (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

Description

This book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds. Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory. Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups. Stein Manifolds and Holomorphic Mappings Preface to the Second Edition Preface to the First Edition Contents Part I: Stein Manifolds Chapter 1: Preliminaries 1.1 Complex Manifolds and Holomorphic Maps 1.2 Examples of Complex Manifolds 1.3 Subvarieties and Complex Spaces 1.4 Holomorphic Fibre Bundles 1.5 Holomorphic Vector Bundles 1.6 The Tangent Bundle 1.7 The Cotangent Bundle and Differential Forms 1.8 Plurisubharmonic Functions and the Levi Form 1.9 Vector Fields, Flows and Foliations 1.10 What is the H-Principle? Chapter 2: Stein Manifolds 2.1 Domains of Holomorphy 2.2 Stein Manifolds and Stein Spaces 2.3 Holomorphic Convexity and the Oka-Weil Theorem 2.4 Embedding Stein Manifolds into Euclidean Spaces 2.5 Characterization by Plurisubharmonic Functions 2.6 Cartan-Serre Theorems A & B 2.7 The -Problem 2.8 Cartan-Oka-Weil Theorem with Parameters Chapter 3: Stein Neighborhoods and Approximation 3.1 Q-Complete Neighborhoods 3.2 Stein Neighborhoods of Stein Subvarieties 3.3 Holomorphic Retractions onto Stein Submanifolds 3.4 A Semiglobal Holomorphic Extension Theorem 3.5 Approximation on Totally Real Submanifolds 3.6 Stein Neighborhoods of Laminated Sets 3.7 Stein Compacts with Totally Real Handles 3.8 A Mergelyan Approximation Theorem 3.9 Strongly Pseudoconvex Handlebodies 3.10 Morse Critical Points of q-Convex Functions 3.11 Critical Levels of a q-Convex Function 3.12 Topological Structure of a Stein Space Chapter 4: Automorphisms of Complex Euclidean Spaces 4.1 Shears 4.2 Automorphisms of C2 4.3 Attracting Basins and Fatou-Bieberbach Domains 4.4 Random Iterations and the Push-Out Method 4.5 Mittag-Leffler Theorem for Entire Maps 4.6 Tame Discrete Sets in Cn 4.7 Unavoidable and Rigid Discrete Sets 4.8 Algorithms for Computing Flows 4.9 The Andersén-Lempert Theory 4.10 The Density Property 4.11 Automorphisms Fixing a Subvariety 4.12 Moving Polynomially Convex Sets 4.13 Moving Totally Real Submanifolds 4.14 Carleman Approximation by Automorphisms 4.15 Automorphisms with Given Jets 4.16 Mittag-Leffler Theorem for Automorphisms of Cn 4.17 Interpolation by Fatou-Bieberbach Maps 4.18 Twisted Holomorphic Embeddings into Cn 4.19 Nonlinearizable Periodic Automorphisms of Cn 4.20 A Non-Runge Fatou-Bieberbach Domain 4.21 A Long C2 Without Holomorphic Functions Part II: Oka Theory Chapter 5: Oka Manifolds 5.1 A Historical Introduction to the Oka Principle 5.2 Cousin Problems and Oka's Theorem 5.3 The Oka-Grauert Principle 5.4 What is an Oka Manifold? 5.5 Basic Properties of Oka manifolds 5.6 Examples of Oka Manifolds 5.7 Cartan Pairs 5.8 A Splitting Lemma 5.9 Gluing Holomorphic Sprays 5.10 Noncritical Strongly Pseudoconvex Extensions 5.11 Proof of Theorem 5.4.4: The Basic Case 5.12 Proof of Theorem 5.4.4: Stratified Fibre Bundles 5.13 Proof of Theorem 5.4.4: The Parametric Case 5.14 Existence Theorems for Holomorphic Sections 5.15 Equivalences Between Oka Properties Chapter 6: Elliptic Complex Geometry and Oka Theory 6.1 Fibre Sprays and Elliptic Submersions 6.2 The Oka Principle for Sections of Stratified Subelliptic Submersions 6.3 Composed and Iterated Sprays 6.4 Examples of Subelliptic Manifolds and Submersions 6.5 Lifting Homotopies to Spray Bundles 6.6 Runge Theorem for Sections of Subelliptic Submersions 6.7 Gluing Holomorphic Sections on C-Pairs 6.8 Complexes of Holomorphic Sections 6.9 C-Strings 6.10 Construction of the Initial Holomorphic Complex 6.11 The Main Modification Lemma 6.12 Proof of Theorems 6.2.2 and 6.6.6 6.13 Relative Oka Principle on 1-Convex Manifolds 6.14 The Oka Principle for Sections of Branched Maps 6.15 Approximation by Algebraic Maps Chapter 7: Flexibility Properties of Complex Manifolds and Holomorphic Maps 7.1 Hierarchy of Holomorphic Flexibility Properties 7.2 Stratified Oka Manifolds and Kummer Surfaces 7.3 Oka Properties of Compact Complex Surfaces 7.4 Oka Maps 7.5 A Homotopy-Theoretic Viewpoint on Oka Theory 7.6 Miscellanea and Open Problems Part III: Applications Chapter 8: Applications of Oka Theory and Its Methods 8.1 Principal Fibre Bundles 8.2 The Oka-Grauert Principle for G-Bundles 8.3 Homomorphisms and Generators of Vector Bundles 8.4 Generators of Coherent Analytic Sheaves 8.5 The Number of Equations Defining a Subvariety 8.6 Elimination of Intersections 8.7 Holomorphic Vaserstein Problem 8.8 Transversality Theorems for Holomorphic Maps 8.9 Singularities of Holomorphic Maps 8.10 Local Sprays of Class A(D) 8.11 Stein Neighborhoods of A(D)-Graphs 8.12 Oka Principle on Strongly Pseudoconvex Domains 8.13 Banach Manifolds of Holomorphic Mappings Chapter 9: Embeddings, Immersions and Submersions 9.1 The H-Principle for Totally Real Immersions and for Complex Submersions 9.2 (Almost) Proper Maps to Euclidean Spaces 9.3 Embedding and Immersing Stein Manifolds into Euclidean Spaces of Minimal Dimension 9.4 Proof of the Relative Embedding Theorem 9.5 Weakly Regular Embeddings and Interpolation 9.6 The Oka Principle for Holomorphic Immersions 9.7 A Splitting Lemma for Biholomorphic Maps 9.8 The Oka Principle for Proper Holomorphic Maps 9.9 Exposing Points of Bordered Riemann Surfaces 9.10 Embedding Bordered Riemann Surfaces in C2 9.11 Infinitely Connected Complex Curves in C2 9.12 Approximation of Holomorphic Submersions 9.13 Noncritical Holomorphic Functions 9.14 The Oka Principle for Holomorphic Submersions 9.15 Closed Holomorphic 1-Forms Without Zeros 9.16 Holomorphic Foliations on Stein Manifolds Chapter 10: Topological Methods in Stein Geometry 10.1 Real Surfaces in Complex Surfaces 10.2 Invariants of Smooth 4-Manifolds 10.3 Lai Indexes and Index Formulas 10.4 Cancelling Pairs of Complex Points 10.5 Applications of the Cancellation Theorem 10.6 The Adjunction Inequality in Kähler Surfaces 10.7 The Adjunction Inequality in Stein Surfaces 10.8 Well Attached Handles 10.9 Stein Structures and the Soft Oka Principle 10.10 The Case dimR X4 10.11 Exotic Stein Structures on Smooth 4-Manifolds References Index