Basic notions and classical results

Abstract

As a preliminary, basic properties of holomorphic functions and complex manifolds are recalled. Beginning with the definitions and characterizations of holomorphic functions, we shall give an overview of the classical theorems in several complex variables, restricting ourselves to extremely important ones for the discussion in later chapters. Most of the materials presented here are contained in well-written textbooks such as Gunning and Rossi (Analytic functions of several complex variables. Prentice-Hall, Inc., Englewood Cliffs, 1965, pp xiv+317), Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Mathematical Library, vol 7. North-Holland Publishing Co., Amsterdam, 1990, pp xii+254), Wells (Differential analysis on complex manifolds, 3rd edn. With a new appendix by Oscar Garcia-Prada. Graduate texts in mathematics, vol 65. Springer, New York, 2008), Grauert and Remmert (Theory of Stein spaces. Translated from the German by Alan Huckleberry. Reprint of the 1979 translation. Classics in mathematics. Springer, Berlin, 2004, pp xxii+255; Coherent analytic sheaves. Grundlehren der Mathematischen Wissenschaften, vol 265. Springer, Berlin, 1984, pp xviii+249) and Noguchi (Analytic function theory of several variables—elements of Oka’s coherence, preprint) (see also Demailly, Analytic methods in algebraic geometry. Surveys of modern mathematics, vol 1. International Press, Somerville/Higher Education Press, Beijing, 2012, pp viii+231) and Ohsawa (Analysis of several complex variables. Translated from the Japanese by Shu Gilbert Nakamura. Translations of mathematical monographs. Iwanami series in modern mathematics, vol 211. American Mathematical Society, Providence, 2002, pp xviii+121), so that only sketchy accounts are given for most of the proofs and historical backgrounds. An exception is Serre’s duality theorem. It will be presented after an article of Laurent-Thiébaut and Leiterer (Some applications of Serre duality in CR manifolds. Nagoya Math J 154:141–156, 1999), since none of the above books contains its proof in full generality.