From Holomorphic Functions to Complex Manifolds
The aim of this book is to give an understandable introduction to the the ory of complex manifolds. With very few exceptions we give complete proofs. Many examples and figures along with quite a few exercises are included. Our intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involved with sheaves, coherence, and higher-dimensional cohomology are avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional co cycles are used. Nevertheless, deep results can be proved, for example the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem. The first chapter deals with holomorphic functions defined in open sub sets of the space en. Many of the well-known properties of holomorphic functions of one variable, such as the Cauchy integral formula or the maxi mum principle, can be applied directly to obtain corresponding properties of holomorphic functions of several variables. Furthermore, certain properties of differentiable functions of several variables, such as the implicit and inverse function theorems, extend easily to holomorphic functions.
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algebraic analytic set analytic subset arbitrary point assume biholomorphic called CC G Cn+1 cocycle codimension cohomology complex manifold connected component continuous converges cuboid decomposition define Definition denote dimension divisor domain of holomorphy domain over Cn element embedded-analytic set exists fiber finite function f functions on G given global Hartogs convex Hartogs figure Hermitian holomorphic function holomorphic map holomorphically convex irreducible components isomorphic Kahler manifold lemma Let f Let G C Cn line bundle linear map f meromorphic function n-dimensional complex manifold open covering open neighborhood open set open subset plurisubharmonic function polydisk polynomial power series projection Proof Proposition Prove pseudopolynomials Reinhardt domain Riemann domain Riemann surface sequence singular Stein manifold strictly plurisubharmonic strongly pseudoconvex subharmonic submanifold tangent vector topological transition functions trivial uniquely determined vanish vector bundle vector space Weierstrass zero set