Introduction to complex analytic geometry. Transl. from the Polish by Maciej Klimek.

*(English)*Zbl 0747.32001
Basel etc.: Birkhäuser Veralg. xiv, 523 p. (1991).

This book is an extended English version of the Polish Edition from 1988. The subject of this book is analytic geometry understood as the geometry of analytic sets (analytic spaces). Except for the last chapter, mostly local problems are investigated and only the complex case is studied. This book is an introduction and its aim is to familiarize the reader with the basic range of problems, using means as elementary as possible. Of great importance is the fact that it is almost self-contained; the author gives the reader access to complete proofs without the need to rely on so-called “well-known” facts. All the necessary properties and theorems have been gathered in the first three preliminary chapters, either with proofs or with references to standard and elementary textbooks.

The first chapter of the book is devoted to the study of the rings of germs of holomorphic functions. The notions of analytic sets and germs are introduced in the second chapter. Some fundamental lemmas (e.g. Rückert’s descriptive lemma) and some consequences (e.g. Hilbert Nullstellensatz) are proved in Chapter III. In the fourth chapter, a study of local structure is followed by an exposition of the basic properties of analytic sets. The Remmert-Stein theorem on removable singularities is proved and analytically constructible sets are presented. The fifth chapter is concerned with holomorhic mappings between analytic sets. Remmert’s proper and open mapping theorems are proved and the Andreotti-Stoll theorem on the structure of finite mappings is presented. Normal spaces and the normalization theorem are the main topic of the sixth chapter. The last chapter is devoted to the relationship between analytic and algebraic geometry.

Though many of the results presented are relatively modern, they are already part of the classical analytic and algebraic geometry.

This monograph may be considered as an excellent introduction to the subject.

The first chapter of the book is devoted to the study of the rings of germs of holomorphic functions. The notions of analytic sets and germs are introduced in the second chapter. Some fundamental lemmas (e.g. Rückert’s descriptive lemma) and some consequences (e.g. Hilbert Nullstellensatz) are proved in Chapter III. In the fourth chapter, a study of local structure is followed by an exposition of the basic properties of analytic sets. The Remmert-Stein theorem on removable singularities is proved and analytically constructible sets are presented. The fifth chapter is concerned with holomorhic mappings between analytic sets. Remmert’s proper and open mapping theorems are proved and the Andreotti-Stoll theorem on the structure of finite mappings is presented. Normal spaces and the normalization theorem are the main topic of the sixth chapter. The last chapter is devoted to the relationship between analytic and algebraic geometry.

Though many of the results presented are relatively modern, they are already part of the classical analytic and algebraic geometry.

This monograph may be considered as an excellent introduction to the subject.

Reviewer: Vasile Brînzănescu (Bucureşti)