多复变量
2009-8
世界图书出版公司
格兰特
207
The present book grew out of introductory lectures on the theory of functionsof several variables. Its intent is to make the reader familiar， by the discussionof examples and special cases， with the most important branches and methodsof this theory， among them， e.g.， the problems of holomorphic continuation，the algebraic treatment of power series， sheaf and cohomology theory， andthe real methods which stem from elliptic partial differential equations. In the first chapter we begin with the definition of holomorphic functionsof several variables， their representation by the Cauchy integral， and theirpower series expansion on Reinhardt domains. It turns out that， in contrastto the theory of a single variable， for n > 2 there exist domains G， Gc Cnwith G G and G =G such that each function holomorphic in G has acontinuation on d. Domains G for which such a d does not exist are calleddomains of holomorphy. In Chapter 2 we give several characterizations ofthese domains of holomorphy （theorem of Cartan-Thullen， Levis problem）.We finally construct the holomorphic hull H（G） for each domain G， that isthe largest （not necessarily schlicht） domain over Cn into which each functionholomorphic on G can be continued. The third chapter presents the Weierstrass formula and the Weierstrasspreparation theorem with applications to the ring of convergent powerseries. It is shown that this ring is a factorization， a Noetherian， and a Henselring. Furthermore we indicate how the obtained algebraic theorems can beapplied to the local investigation of analytic sets. One achieves deep resultsin this connection by using sheaf theory， the basic concepts of which arediscussed in the fourth chapter. In Chapter V we introduce complex manifoldsand give several examples. We also examine the different closures of C andthe effects of modifications on complex manifolds.
The third chapter presents the Weierstrass formula and the Weierstrasspreparation theorem with applications to the ring of convergent powerseries. It is shown that this ring is a factorization， a Noetherian， and a Henselring. Furthermore we indicate how the obtained algebraic theorems can beapplied to the local investigation of analytic sets. One achieves deep resultsin this connection by using sheaf theory， the basic concepts of which arediscussed in the fourth chapter. In Chapter V we introduce complex manifoldsand give several examples. We also examine the different closures of C andthe effects of modifications on complex manifolds.
Chapter Ⅰ Holomorphic Functions 1 Power Series 2 Complex Differentiable Functions 3 The Cauchy Integral 4 Identity Theorems 5 Expansion in Reinhardt Domains 6 Real and Complex Differentiability 7 Holomorphic MappingsChapter Ⅱ Domains of Holomorphy 1 The Continuity Theorem 2 Pseudoconvexity 3 Holomorphic Convexity 4 The Thullen Theorem 5 Holomorphically Convex Domains: 6 Examples 7 Riemann Domains over Cn 8 Holomorphic HullsChapter Ⅲ The Weierstrass Preparation Theorem 1 The Algebra of Power Series 2 The Weierstrass Formula 3 Convergent Power Series 4 Prime Factorization 5 Further Consequences (Hensel Rings, Noetherian Rings) 6 Analytic SetsChapter Ⅳ Sheaf Theory 1 Sheaves of Sets 2 Sheaves with Algebraic Structure 3 Analytic Sheaf Morphisms 4 Coherent SheavesChapter Ⅴ Complex Manifolds 1 Complex Ringed Spaces 2 Function Theory on Complex Manifolds 3 Examples of Complex Manifolds 4 Closures of CnChapter Ⅵ Cohomology Theory 1 Flabby Cohomology 2 The Cech Cohomology 3 Double Complexes 4 The Cohomology Sequence 5 Main Theorem on Stein ManifoldsChapter Ⅶ Real Methods 1 Tangential Vectors 2 Differential Forms on Complex Manifolds 3 Cauchy Integrals 4 Dolbeault's Lemma 5 Fine Sheaves (Theorems of Dolbeault and de Rham)List of symbolsBibliographyIndex
sheaf and cohomology theory， andthe real methods which stem from elliptic partial differential equations.
多的就是比单的要有难度，需要多下功夫好好地读这本书