Analytic Function Theory of Several Variables: Elements of by Junjiro Noguchi

By Junjiro Noguchi

The function of this booklet is to give the classical analytic functionality conception of numerous variables as a typical topic in a process arithmetic after studying the trouble-free fabrics (sets, normal topology, algebra, one advanced variable). This comprises the fundamental elements of Grauert–Remmert's volumes, GL227(236) (Theory of Stein spaces) and GL265 (Coherent analytic sheaves) with a reducing of the extent for amateur graduate scholars (here, Grauert's direct picture theorem is proscribed to the case of finite maps).
The center of the speculation is "Oka's Coherence", came upon and proved via Kiyoshi Oka. it truly is essential, not just within the research of advanced research and intricate geometry, but additionally in a wide region of contemporary arithmetic. during this e-book, simply after an introductory bankruptcy on holomorphic features (Chap. 1), we end up Oka's First Coherence Theorem for holomorphic capabilities in Chap. 2. This defines a distinct personality of the e-book in comparison with different books in this topic, during which the thought of coherence seems to be a lot later.
The current e-book, along with 9 chapters, provides whole remedies of the next goods: Coherence of sheaves of holomorphic capabilities (Chap. 2); Oka–Cartan's primary Theorem (Chap. 4); Coherence of excellent sheaves of complicated analytic subsets (Chap. 6); Coherence of the normalization sheaves of advanced areas (Chap. 6); Grauert's Finiteness Theorem (Chaps. 7, 8); Oka's Theorem for Riemann domain names (Chap. 8). The theories of sheaf cohomology and domain names of holomorphy also are provided (Chaps. three, 5). bankruptcy 6 offers with the speculation of complicated analytic subsets. bankruptcy eight is dedicated to the functions of previously got effects, proving Cartan–Serre's Theorem and Kodaira's Embedding Theorem. In Chap. nine, we speak about the ancient improvement of "Coherence".
It is hard to discover a booklet at this point that treats the entire above topics in a very self-contained demeanour. within the current quantity, a few classical proofs are enhanced and simplified, in order that the contents are simply obtainable for starting graduate students.

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