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An introduction to CR structures. (English) Zbl 0712.32001
Mathematical Surveys and Monographs, 32. Providence, RI: American Mathematical Society (AMS). x, 237 p. $ 74.00 (1990).
Any two real analytic curves in \({\mathbb{C}}\) are locally biholomorphically equivalent. After Poincaré had shown that the analogous result does not hold in \({\mathbb{C}}^ 2\), a natural question was raised of finding invariants that distinguish one real hypersurface from another. This problem was solved by E. Cartan in 1932. Fourty years later Moser gave another solution and generalized it to dimensions greater than 2 in his joint work with Chern. One of the basic points was the observation that the restriction of a holomorphic function to a hypersurface must satisfy certain partial differential equations. This leads to the introduction of CR structures which are the subject of the book.
It must be pointed out that most of the book is devoted to the exposition of the important works of Cartan and Chern-Moser. The author provides the necessary background in the first chapter in detail and with numerous useful examples and exercises. He also pays much attention to geometric aspects of the theory and to the examination of relations between Cartan’s invariants and those of Moser. The last chapter contains the results concerning the nonsolvability of the Levy operator. The book contains also historical notes and a rather brief, to reviewer’s mind, list of references.
Reviewer: A.Russakovskii

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32V40 Real submanifolds in complex manifolds