Stein Manifolds and Holomorphic Mappings [electronic resource] : The Homotopy Principle in Complex Analysis /
Forstnerič, Franc.author.(orcid)0000-0002-7975-0212https://orcid.org/0000-0002-7975-0212aut
The main theme of this book is the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds. The book contains the first complete account of Oka-Grauert theory and its modern extensions, initiated by Mikhail Gromov and developed in the last decade by the author and his collaborators. Included is the first systematic presentation of the theory of holomorphic automorphisms of complex Euclidean spaces, a survey on Stein neighborhoods, connections between the geometry of Stein surfaces and Seiberg-Witten theory, and a wide variety of applications ranging from classical to contemporary.
Description based upon print version of record.
Stein Manifolds and Holomorphic Mappings; Preface; Contents; 1 Preliminaries; 1.1 Complex Manifolds and Holomorphic Mappings; 1.2 Examples of Complex Manifolds; 1.3 Subvarieties and Complex Spaces; 1.4 Holomorphic Fiber 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 Jet Bundles, Holonomic Sections and the Homotopy Principle; 2 Stein Manifolds; 2.1 Domains of Holomorphy; 2.2 Stein Manifolds and Stein Spaces
2.3 Characterization by Plurisubharmonic Functions2.4 Cartan-Serre Theorems A & B; 2.5 The partial-Problem; 3 Stein Neighborhoods and Holomorphic 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 Totally Real Submanifolds; 3.6 Stein Neighborhoods of Certain Laminated Sets; 3.7 Stein Compacts with Totally Real Handles; 3.8 Thin Strongly Pseudoconvex Handlebodies; 3.9 Morse Critical Points of q-Convex Functions
3.10 Crossing a Critical Level of a q-Convex Function3.11 The Topological Structure of a Stein Space; 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 Vector Fields; 4.9 The Andersén-Lempert Theorem; 4.10 The Density Property; 4.11 Automorphisms Fixing a Subvariety; 4.12 Moving Polynomially Convex Sets
4.13 Moving Totally Real Submanifolds4.14 Controlling Unbounded Curves; 4.15 Automorphisms with Given Jets; 4.16 A Mittag-Leffler Theorem for Automorphisms of Cn; 4.17 Interpolation by Fatou-Bieberbach Maps; 4.18 Twisted Holomorphic Embeddings Ck to Cn; 4.19 Nonlinearizable Periodic Automorphisms of Cn; 4.20 Non-Runge Fatou-Bieberbach Domains and Long Cn's; 4.21 Serre's Problem on Stein Bundles; 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 Examples of Oka Manifolds
5.6 An Application of Michael's Selection Theorem5.7 Cartan Pairs; 5.8 A Splitting Lemma; 5.9 Gluing Local Holomorphic Sprays; 5.10 Noncritical Strongly Pseudoconvex Extensions; 5.11 Proof of the Main Theorem: The Basic Case; 5.12 Proof of the Main Theorem: Stratified Fiber Bundles; 5.13 Proof of the Main Theorem: The Parametric Case; 5.14 Existence Theorems for Holomorphic Sections; 5.15 Equivalences Between Oka Properties; 5.16 Open Problems; 6 Elliptic Complex Geometry and Oka Principle; 6.1 Holomorphic Fiber-Sprays and Elliptic Submersions; 6.2 Gromov's Oka Principle
6.3 Composed and Iterated Sprays
Includes bibliographical references and index.
The main theme of this book is the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds. The book contains the first complete account of Oka-Grauert theory and its modern extensions, initiated by Mikhail Gromov and developed in the last decade by the author and his collaborators. Included is the first systematic presentation of the theory of holomorphic automorphisms of complex Euclidean spaces, a survey on Stein neighborhoods, connections between the geometry of Stein surfaces and Seiberg-Witten theory, and a wide variety of applications ranging from classical to contemporary.
Functions of real variables.
Functions of complex variables.
Real Functions.
Functions of a Complex Variable.
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