03699nam a22004455i 4500001001800000003000900018005001700027007001500044008004100059020001800100024003500118050001200153072001700165072002300182072001600205100007900221245009600300250001800396264007200414300006400486336002600550337002600576338003600602347002400638490002900662505020500691520200900896650003602905650001302941650001402954650005102968650001303019650001403032710003403046773002603080776003603106776003603142830002903178856004603207978-981-97-2056-9DE-He21320251030112640.0cr nn 008mamaa240720s2024    si |    s    |||| 0|eng d  a97898197205697 a10.1007/978-981-97-2056-92doi 4aQA331.7 7aPBKD2bicssc 7aMAT0340002bisacsh 7aPBKD2thema1 aNoguchi, Junjiro.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut10aBasic Oka Theory in Several Complex Variablesh[electronic resource] /cby Junjiro Noguchi.  a1st ed. 2024. 1aSingapore :bSpringer Nature Singapore :bImprint: Springer,c2024.  aXVI, 221 p. 22 illus., 5 illus. in color.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aUniversitext,x2191-66750 a1 Holomorphic Functions -- 2 Coherent Sheaves and Oka’s Joku-Iko Principle -- 3 Domains of Holomorphy -- 4 Pseudoconvex Domains I — Problem and Reduction -- 5 Pseudoconvex Domains II — Solution.  aThis book provides a new, comprehensive, and self-contained account of Oka theory as an introduction to function theory of several complex variables, mainly concerned with the Three Big Problems (Approximation, Cousin, Pseudoconvexity) that were solved by Kiyoshi Oka and form the basics of the theory. The purpose of the volume is to serve as a textbook in lecture courses right after complex function theory of one variable. The presentation aims to be readable and enjoyable both for those who are beginners in mathematics and for researchers interested in complex analysis in several variables and complex geometry. The nature of the present book is evinced by its approach following Oka’s unpublished five papers of 1943 with his guiding methodological principle termed the “Joku-Iko Principle”, where historically the Pseudoconvexity Problem (Hartogs, Levi) was first solved in all dimensions, even for unramified Riemann domains as well. The method that is used in the book is elementary and direct, not relying on the cohomology theory of sheaves nor on the L2-∂-bar method, but yet reaches the core of the theory with the complete proofs. Two proofs for Levi’s Problem are provided: One is Oka’s original with the Fredholm integral equation of the second kind combined with the Joku-Iko Principle, and the other is Grauert’s by the well-known “bumping-method” with L. Schwartz’s Fredholm theorem, of which a self-contained, rather simple and short proof is given. The comparison of them should be interesting even for specialists. In addition to the Three Big Problems, other basic material is dealt with, such as Poincaré’s non-biholomorphism between balls and polydisks, the Cartan–Thullen theorem on holomorphic convexity, Hartogs’ separate analyticity, Bochner’s tube theorem, analytic interpolation, and others. It is valuable for students and researchers alike to look into the original works of Kiyoshi Oka, which are not easy to find in books or monographs. 0aFunctions of complex variables. 0aAlgebra. 0aTopology.14aSeveral Complex Variables and Analytic Spaces.24aAlgebra.24aTopology.2 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978981972055208iPrinted edition:z9789819720576 0aUniversitext,x2191-667540uhttps://doi.org/10.1007/978-981-97-2056-9