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, Englewood Cliffs, xiv+317 pp, 1965), Hormander (An introduction to complex analysis in several variables. North-Holland Mathematical Library, vol 7, 3rd edn. North-Holland, Amsterdam, xii+254 pp, 1990), Wells and Raymond (Differential analysis on complex manifolds, Third edition. 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, xxii+255 pp, 2004), Grauert and Remmert (Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften, vol 265. Springer, Berlin, xviii+249 pp, 1984) and Noguchi (Analytic function theory of several variables —Elements of Oka’s coherence, preprint (translated from Japanese)) (see also Demailly, Analytic methods in algebraic geometry. Surveys of modern mathematics, vol 1. International Press, Somerville; Higher Education Press, Beijing, viii+231 pp, 2012 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, xviii+121 pp, 2002), 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-Thiebaut and Leiterer (Nagoya Math J 154:141–156, 1999), since none of the above books contains its proof in full generality.