Basic Oka Theory in Several Complex Variables

• Record Nr.: UNINA9910874688603321
• Autore: Noguchi Junjiro
• Pubbl/distr/stampa: Singapore : , : Springer, , 2024; ©2024
• ISBN: 9789819720569; 9789819720552
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (232 pages)
• Collana: Universitext Series
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Preface -- Contents -- Conventions -- Chapter 1 Holomorphic Functions -- 1.1 Holomorphic Functions of Several Variables -- 1.1.1 Open Balls and Polydisks of Cn -- 1.1.2 Definition of Holomorphic Functions -- 1.1.3 Sequences and Series of Functions -- 1.1.4 Power Series of Several Variables -- 1.1.5 Elementary Properties of Holomorphic Functions of Several Variables -- 1.2 Analytic Continuation and Hartogs' Phenomenon -- 1.3 Runge Approximation on Convex Cylinder Domains -- 1.3.1 Cousin Integral -- 1.4 Implicit and Inverse Function Theorems -- 1.5 Analytic Subsets -- Exercises -- Chapter 2 Coherent Sheaves and Oka's Joku-Iko Principle -- 2.1 Notion of Analytic Sheaves -- 2.1.1 Definitions of Rings and Modules -- 2.1.2 Analytic Sheaves -- 2.2 Coherent Sheaves -- 2.2.1 Locally Finite Sheaves -- 2.2.2 Coherent Sheaves -- 2.3 Oka's First Coherence Theorem -- 2.3.1 Weierstrass' Preparation Theorem -- 2.3.2 Oka's First Coherence Theorem -- 2.3.3 Coherence of Ideal Sheaves of Complex Submanifolds -- 2.4 Cartan's Merging Lemma -- 2.4.1 Matrices and Matrix-Valued Functions -- 2.4.2 Cartan's Matrix Decomposition -- 2.4.3 Cartan's Merging Lemma -- 2.5 Oka's Joku-Iko Principle -- 2.5.1 Oka Syzygy -- 2.5.2 Oka Extension of the Joku-Iko Principle -- Exercises -- Chapter 3 Domains of Holomorphy -- 3.1 Definitions and Elementary Properties -- 3.1.1 Relatively Compact Hull -- 3.1.2 Domain of Holomorphy and Holomorphic Convexity -- 3.2 Cartan-; Thullen Theorem -- 3.3 Analytic Polyhedron and Oka-Weil Approximation -- 3.3.1 Analytic Polyhedron -- 3.3.2 Oka-Weil Approximation Theorem -- 3.3.3 Runge Approximation Theorem (One Variable) -- 3.4 Cousin Problem -- 3.4.1 Cousin I Problem -- 3.4.2 Continuous Cousin Problem -- 3.4.3 Cousin I Problem-continued -- 3.4.4 Hartogs Extension over a Compact Subset -- 3.4.5 Mittag-Leffler Theorem (One Variable).; 3.4.6 Cousin II Problem and Oka Principle -- 3.4.7 Weierstrass' Theorem (One Variable) -- 3.4.8 ¯∂-Equation -- 3.5 Analytic Interpolation Problem -- 3.6 Unramified Domains over Cn -- 3.7 Stein Domains over Cn -- 3.8 Supplement: Ideal Boundary -- Exercises -- Chapter 4 Pseudoconvex Domains I - Problem and Reduction -- 4.1 Plurisubharmonic Functions -- 4.1.1 Subharmonic Functions (One Variable) -- 4.1.2 Plurisubharmonic Functions -- 4.1.3 Smoothing -- 4.2 Hartogs' Separate Analyticity -- 4.2.1 Baire Category Theorem -- 4.2.2 Separate Analyticity -- 4.3 Pseudoconvexity -- 4.3.1 Pseudoconvexity Problem -- 4.3.2 Bochner's Tube Theorem -- 4.3.3 Pseudoconvex Boundary -- 4.3.4 Levi Pseudoconvexity -- 4.3.5 Strongly Pseudoconvex Boundary Points and Stein Domains -- Exercises -- Chapter 5 Pseudoconvex Domains II - Solution -- 5.1 The Oka Extension with Estimate -- 5.1.1 Preparation from Topological Vector Spaces -- 5.1.2 The Oka Extension with Estimate -- 5.2 Strongly Pseudoconvex Domains -- 5.2.1 Oka's Method -- 5.2.2 Grauert's Method -- 5.3 Oka's Pseudoconvexity Theorem -- Exercises -- Afterword- Historical Comments -- References -- Index -- Symbols.