Analytic Functions of Several Complex VariablesThe theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces. |
Contents
| 1 | |
Chapter IILocal Rings of Holomorphic Functions | 65 |
Chapter IIIVarieties | 93 |
Chapter IVAnalytic Sheaves | 118 |
Chapter VAnalytic Spaces | 147 |
Chapter VICohomology Theory | 172 |
Chapter VIIStein Spaces Geometric Theory | 208 |
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Analytic Functions of Several Complex Variables Robert Clifford Gunning,Hugo Rossi No preview available |
Common terms and phrases
a₁ abelian groups analytic cover analytic polyhedron analytic sheaves analytic space biholomorphic chain of syzygies coherent analytic sheaf coherent sheaf cohomology compact set compact subset complex manifold complex variables connected converges coordinates Corollary defined Definition dense differential form element exact sequence əzi f₁ finite follows Frechet space function f function holomorphic g₁ h₁ hence holo holomorphic functions holomorphic mapping holomorphically convex ideal induction integral irreducible branch isomorphism K₁ kernel lemma Let f Math matrix modules morphic nonsingular open neighborhood open set open subset polydisc presheaf Proof proposition prove pseudoconvex pure dimension R-modules relatively compact Riemann domain sheaf homomorphism sheaf of germs sheaf of rings Stein manifold Stein space submanifold subvariety Suppose syzygies theorem topology U₁ V₁ V₂ vanishes variety vector W₁ Weierstrass polynomial X₁ z₁ zero