LEADER 04206nam 22003973 450 001 9910874688603321 005 20240723080219.0 010 $a9789819720569$b(electronic bk.) 010 $z9789819720552 035 $a(MiAaPQ)EBC31538791 035 $a(Au-PeEL)EBL31538791 035 $a(CKB)33101458600041 035 $a(EXLCZ)9933101458600041 100 $a20240723d2024 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aBasic Oka Theory in Several Complex Variables 205 $a1st ed. 210 1$aSingapore :$cSpringer,$d2024. 210 4$d©2024. 215 $a1 online resource (232 pages) 225 1 $aUniversitext Series 311 08$aPrint version: Noguchi, Junjiro Basic Oka Theory in Several Complex Variables Singapore : Springer,c2024 9789819720552 327 $aIntro -- 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). 327 $a3.4.6 Cousin II Problem and Oka Principle -- 3.4.7 Weierstrass' Theorem (One Variable) -- 3.4.8 ¯?-Equation -- 3.5 Analytic Interpolation Problem -- 3.6 Unramified Domains over Cn -- 3.7 Stein Domains over Cn -- 3.8 Supplement: Ideal Boundary -- Exercises -- Chapter 4 Pseudoconvex Domains I - Problem and Reduction -- 4.1 Plurisubharmonic Functions -- 4.1.1 Subharmonic Functions (One Variable) -- 4.1.2 Plurisubharmonic Functions -- 4.1.3 Smoothing -- 4.2 Hartogs' Separate Analyticity -- 4.2.1 Baire Category Theorem -- 4.2.2 Separate Analyticity -- 4.3 Pseudoconvexity -- 4.3.1 Pseudoconvexity Problem -- 4.3.2 Bochner's Tube Theorem -- 4.3.3 Pseudoconvex Boundary -- 4.3.4 Levi Pseudoconvexity -- 4.3.5 Strongly Pseudoconvex Boundary Points and Stein Domains -- Exercises -- Chapter 5 Pseudoconvex Domains II - Solution -- 5.1 The Oka Extension with Estimate -- 5.1.1 Preparation from Topological Vector Spaces -- 5.1.2 The Oka Extension with Estimate -- 5.2 Strongly Pseudoconvex Domains -- 5.2.1 Oka's Method -- 5.2.2 Grauert's Method -- 5.3 Oka's Pseudoconvexity Theorem -- Exercises -- Afterword- Historical Comments -- References -- Index -- Symbols. 410 0$aUniversitext Series 700 $aNoguchi$b Junjiro$059671 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910874688603321 996 $aBasic Oka Theory in Several Complex Variables$94183838 997 $aUNINA