Open algebraic surfaces are, by definition, algebraic surfaces that are not necessarily complete. Usually, an open algebraic surface is understood as a Zariski open subset of a projective algebraic surfaces. Important examples of open surfaces are provided by the class of affine surfaces, and those can be interpreted as complements of hypersurface sections of projective surfaces. While the theory of projective surfaces has a long history, and reached a high-developed state of art, culminating in the beautiful Enriques-Kodaira classification of such surfaces, the theory of open algebraic surfaces is comparatively young. In fact, there was limited knowledge of the geometric or topological structure of open algebraic surfaces until the early 1970’s when C. P. Ramanujam, S. S. Abhyankar, T. T. Moh, M. Nagata, and others began to study those surfaces systematically. These researchers developed the right methods for overcoming the crucial difficulty of determining the boundary behavior of affine varieties and their subvarieties, and another break-through in the study of open algebraic surfaces occurred in the late 1970’s, when S. Iitaka introduced the notion of logarithmic Kodaira dimension of an algebraic variety and indicated the possibility of classifying non-complete algebraic varieties via logarithmic Kodaira dimension in a way parallel to the classification of projective varieties by means of this invariant. Specifically, in the case of surfaces, this suggested to use the Enriques-Kodaira classification of projective surfaces also in the framework of open algebraic surfaces.
The book under review is the first comprehensive research monograph (and textbook, too) on the theory of open surfaces. The author himself, who has contributed a great deal to the recent developments in this field, and is one of the leading experts in this area of research, had published a first research monograph on non-complete algebraic surfaces some twenty years ago [cf.: M. Miyanishi, “Non-complete algebraic surfaces”. Lect. Notes Math. 857 (1981; Zbl 0456.14018)], when the theory was still at an early stage of its development. Now he presents an up-to-date text that covers the entire theory of open surfaces, as it has been developed so far, in a systematic, comprehensive and very detailed exposition.
The book consists of three chapters, each of which is subdivided into several sections. Chapter 1 is entitled “Complete algebraic surfaces” and recalls the Enriques-Kodaira classification of complete algebraic surfaces on 58 pages. This includes the general fundamental theorems for algebraic surfaces, the geometry of ruled surfaces, the structure of elliptic and quasi-elliptic fibrations, the complete Enriques-Kodaira classification, and the geometry of quotient singularities in normal complete surfaces.
The author gives all the details and full proofs, which makes this chapter into another excellent text on the classical Enriques-Kodaira classification of complete surfaces. However, his strategic goal, with including this well-known material, was to achieve a full comparison between the two theories, as needed later on, and to cover the case of positive characteristic as well.
Chapter 2, entitled “Open algebraic surfaces”, develops the general theory of open surfaces in full detail. This chapter comes with six sections:
1. Kodaira dimensions (general theory);
2. Algebraic surfaces with logarithmic Kodaira dimension;
3. Theory of peeling;
4. Log-projective surfaces and minimal models;
5. Log-del Pezzo surfaces of rank one;
6. Structure theorems for open algebraic surfaces with Kodaira dimension 0 or 1.
Obviously, this chapter is the core of the book.
Chapter 3 turns to the important special case of affine algebraic surfaces and covers a wealth of very recent results in this area. Most of the material appears for the first time in a monograph. This chapter consists of four sections:
1. Fibrations (especially affine fibrations);
2. Algebraic characterizations of the affine plane (à la M. Miyanishi and R. Swan);
3. The theorem of Abhyankar-Moh-Suzuki and the Theorem of Lin-Zaidenberg;
4. Homology planes (outline of the classification theory).
The text is enhanced by a nearly complete bibliography, which is often and pointed referred to, throughout the book. The tremendous value of this research monograph (with textbook character) can barely be overestimated. The author has filled a painful gap in the vast literature on algebraic suaces, and his book will certainly serve as “the” source book on non-complete a algebraic surfaces in the future. The text is written in just as brilliant a manner as his textbook “Algebraic geometry” (1994; Zbl 0808.14001) which may be regarded as probably the best introduction to the more special book under review.