For the past 10 years there has been a bumper crop of books in algebraic geometry which deal with the classification of algebraic surfaces [A. Beauville, Complex algebraic surfaces (French), Asterisque, 54, Soc. Math. France, Paris, 1978; MR0485887; English translation, Cambridge Univ. Press, Cambridge, 1983; MR0732439; L. Badescu, Algebraic surfaces (Romanian), Ed. Acad. R. S. Romania, Bucharest, 1981; MR0633149; H. Kurke, Lectures on algebraic surfaces (German), Teubner, Leipzig, 1982; MR0680403; P. A. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978; MR0507725]. However, all of them served only as an introduction to the subject. The book under review goes far beyond this goal. For the first time, the Enriques classification of compact algebraic surfaces is complemented by the Kodaira classification of compact complex surfaces; also, many of the most recent results on surfaces of general type and surfaces of type K3 are fully represented. The title of the book emphasises the role of transcendental methods in the theory. This choice is certainly necessary for the treatment of nonalgebraic complex surfaces. It also makes this book more accessible to nonspecialists in algebraic geometry. The reader interested in learning the projective technique and the beauty of classic Italian style theory of algebraic surfaces should look for another book (those of Griffiths-Harris or Beauville give much better introductions into this subject). As an illustration of the spirit of this book, we should mention the absence in it of an example of a cubic surface with its 27 lines, the symbol of the theory of algebraic surfaces for more than a century. The authors did not plan to write an encyclopedia of algebraic surfaces; many topics are intentionally left aside. Among them are the theory of rational surfaces (including Cremona transformations), topology of surfaces, automorphisms and moduli of surfaces. The contents of the chapters are the following. The first 3 chapters (``Preliminaries'', ``Curves on surfaces'', ``Mappings of surfaces'') contain the necessary general technique. Among the nonstandard tools included here are Yau's results on Kahler-Einstein metrics, the arithmetic of integral quadratic forms, rational singularities, stable curve fibrations and Iitaka's conjecture (C2,1). Chapter IV, ``Some general properties of surfaces'', collects several general theorems about complex surfaces (the signature theorem, relations between topological and algebraic invariants, projectivity criteria and vanishing theorems are included). Chapter V, ``Examples'', contains not only examples (such as complete intersections, projective plane, tori, Hopf surfaces, Inoue surfaces, Hilbert modular surfaces and Enriques surfaces), it contains also most of Kodaira's theory of elliptic surfaces. Chapter VI, ``The Enriques-Kodaira classification'', is certainly the heart of the book. The authors' approach to the classification is new. It is based on Iitaka's conjecture (C2,1) proved earlier, in Chapter III. This chapter ends with a discussion of analytic deformation of surfaces. Chapter VII, ``Surfaces of general type'', contains two principal results: Miyaoka's proof of the inequality c21≤3c2 and the Kodaira-Bombieri analysis of pluricanonical maps of surfaces of general type. The chapter ends with the discussion of known results about the construction of surfaces with given values of numerical invariants (the list of all known surfaces with pg=0 is included). The last chapter, ``K3-surfaces and Enriques surfaces'', is probably the best chapter of the book. Here, the authors collect (with full proofs) all the most recent results on the periods of K3-surfaces: the global Torelli theorem, the surjectivity theorem and the existence of a Kahler metric on them. The classic theory of Enriques surfaces is supplemented with a new theory of its periods (defined as the periods of its K3-cover). The book begins with a historical account of the theory of surfaces. Also, some of the chapters contain bibliographical remarks. This is the most vulnerable part of the book. Quite understandably, it is impossible to give full credit to all the people who have contributed to the subject. However, it is difficult to justify the absence of mentioning the role of the book Algebraic surfaces by Shafarevich and his students in the historical account [I. R. Shafarevich et al. , Trudy Mat. Inst. Steklov. 75 (1965), 1–125; MR0190143; English translation, Amer. Math. Soc., Providence, R. I., 1967; MR0215850]. This book was the first one to put the classic theory of algebraic surfaces (the case of nonalgebraic surfaces was not considered) on a modern footing and appeared before or at the same time as most of Kodaira's papers on classification of complex surfaces were published. The authors even fail to mention that ``the basic idea: first the classification according to Kodaira dimension…'' (cited by the authors), is due to Shafarevich (cf. Beauville's book [op. cit.] for the corresponding remark): neither Enriques nor Kodaira used it for the basis of their classification. One could also mention other small inaccuracies and unfairnesses in bibliographical remarks.
Reviewer: Dolgachev, I. [form MathSciNet]