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**First steps in several complex variables: Reinhardt domains.**
*(English)*
Zbl 1148.32001

EMS Textbooks in Mathematics. Zürich: European Mathematical Society (ISBN 978-3-03719-049-4/hbk). viii, 359 p. (2008).

The class of Reinhardt domains (i.e., the domains in \({\mathbb C}^n\) invariant under any biholomorphism of the form \(f(z_1, \dots, z_n) = (e^{i \theta_1} z_1, \dots, e^{i \theta_n} z_n)\)) always played a crucial role in the theory of several complex variables. There are at least two simple reasons for that: (a) any holomorphic function, defined on a neighborhood of \(0\), has a Reinhardt domain as domain of convergence; (b) since most of the geometric invariants of a Reinhardt domain depend only on the moduli \(| z_i| \) of the coordinates of its points, many general questions reduce in the context of Reinhardt domains to much simpler problems concerning just real functions of real variables. It is difficult to find other families of complex domains, for which so many interesting information are either known or promptly computable on demand. Due to this, the class of Reinhardt domains is quite often one of the very first places, where one looks for examples or counterexamples to check properties and conjectures.

As the authors explain in the Introduction, this book has been written with the aim of being useful as a textbook for beginning graduate students and as source book for lectures and seminars. The reviewer would like to add that it can be considered also as a good aid for researchers that want to get a fairly complete panorama on what is known on Reinhardt domains.

The book is divided into four chapters. In Ch. 1, the basic topics of several complex variables are covered (domains of convergence, domains of holomorphy, plurisubharmonic functions, pseudoconvexity, Levi problem, etc.): Each topic is at first treated in generality and then analyzed in the context of Reinhardt domains. In Ch. 2, the authors consider the classification problem of automorphism groups and the equivalence classes of domains with given automorphism group: Again, they first review general results on the automorphism groups of domains in \({\mathbb C}^n\) and secondly, they present what is known on this regard for Reinhardt domains. Ch. 3 is devoted to the characterizations of Reinhardt domains which are \(S\)-domains of holomorphy, where \(S\) varies amongst many natural Fréchet spaces of holomorphic functions. In Ch. 4, the authors review all main definitions and properties of biholomorphically invariant pseudodistances and pseudometrics (like e.g., Kobayashi and Carathéodory pseudometrics) and of other holomorphically contractible families of functions. For all of these objects, many examples and most of the known properties in the context of Reinhardt domains are given.

The book is very nicely written and it is enriched by many useful exercises and various suggestions for further investigations.

As the authors explain in the Introduction, this book has been written with the aim of being useful as a textbook for beginning graduate students and as source book for lectures and seminars. The reviewer would like to add that it can be considered also as a good aid for researchers that want to get a fairly complete panorama on what is known on Reinhardt domains.

The book is divided into four chapters. In Ch. 1, the basic topics of several complex variables are covered (domains of convergence, domains of holomorphy, plurisubharmonic functions, pseudoconvexity, Levi problem, etc.): Each topic is at first treated in generality and then analyzed in the context of Reinhardt domains. In Ch. 2, the authors consider the classification problem of automorphism groups and the equivalence classes of domains with given automorphism group: Again, they first review general results on the automorphism groups of domains in \({\mathbb C}^n\) and secondly, they present what is known on this regard for Reinhardt domains. Ch. 3 is devoted to the characterizations of Reinhardt domains which are \(S\)-domains of holomorphy, where \(S\) varies amongst many natural Fréchet spaces of holomorphic functions. In Ch. 4, the authors review all main definitions and properties of biholomorphically invariant pseudodistances and pseudometrics (like e.g., Kobayashi and Carathéodory pseudometrics) and of other holomorphically contractible families of functions. For all of these objects, many examples and most of the known properties in the context of Reinhardt domains are given.

The book is very nicely written and it is enriched by many useful exercises and various suggestions for further investigations.

Reviewer: Andrea Spiro (Camerino)

### MSC:

32-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |

32A05 | Power series, series of functions of several complex variables |

32A10 | Holomorphic functions of several complex variables |

32D05 | Domains of holomorphy |

32E05 | Holomorphically convex complex spaces, reduction theory |

32F17 | Other notions of convexity in relation to several complex variables |

32F45 | Invariant metrics and pseudodistances in several complex variables |

32M05 | Complex Lie groups, group actions on complex spaces |