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Complex analysis and CR geometry. (English) Zbl 1160.32001

University Lecture Series 43. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4442-7/pbk). viii, 200 p. (2008).
As mentioned by the author in the Introduction, this book is designed for two purposes: to provide a short but fairly complete exposition of the basics of the theory of holomorphic functions of several complex variables and of CR functions and, at the same time, to gradually bring the reader to a solid understanding of several topics of current research in those fields. In our opinion, both aims are outstandingly achieved with a very well balanced choice of topics and a constantly clear and rigorous exposition.
The book is divided in three chapters. In the first one, the reader can find an introduction to the theory of several complex variables, focused on the solution of the Levi problem in terms of Khon’s method of solutions of the \(\bar \partial\)-Neumann problem, together with a presentation of some advanced topics, like e.g., regularity properties of the \(\bar \partial\)-operator at a weakly \(q\)-pseudoconvex boundary. In the second chapter, there is a presentation of basic theorems on involutive and non-involutive subbundles of the tangent space (like e.g., the Frobenius and Darboux-Frobenius theorems), results on subellipticity and hypo-subellipticity of systems of vector fields and some other results on the “orbits” of the bundles generated by non-integrable distributions. The material of this second chapter can be considered as preparatory to the last chapter (chapter 3), where the author introduces the notion of CR manifolds and CR functions and discusses several techniques and new results on the extendibility of CR functions. The exposition is mainly concentrated on the applications of properties of conormal bundles and of the lifts of attached holomorphic discs of a CR manifold. Several new proofs and refinements of classical results, together with suggestions for further studies, are presented.
This short stimulating book will be very likely appreciated not only by graduate students in complex analysis and theory of CR functions, but also by experienced researchers interested in those areas.

MSC:

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
32Fxx Geometric convexity in several complex variables
32Txx Pseudoconvex domains
32Vxx CR manifolds
32Wxx Differential operators in several variables
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