This textbook on algebraic geometry may be regarded as a highly welcome complement to others which have appeared since A. Grothendieck's rigorous foundation of algebraic geometry from the ``scheme-theoretic'' point of view. More than those others, this book reflects the intimate connection of algebraic geometry with complex analysis as one of the roots of algebraic geometry, in addition to emphasizing the detailed study of concrete classical and nonclassical geometric questions. Therefore, the ground field is always the field of complex numbers, and considerations are limited to projective varieties. As the authors point out in their preface, the methodological principles of this book are the following: (1) The general machinery and techniques should be developed only insofar as they are necessary to handle some concrete geometric questions and special classes of algebraic varieties. (2) There should be a balance between the general theory and the study of examples. (3) The book should be completely self-contained to facilitate the study and to avoid confusing cross-references, which are so typical of the literature in algebraic geometry. According to these principles and the chosen approach the book begins with a very detailed exposition of the fundamental techniques and results of complex manifold theory, especially the Weierstrass theorems and their corollaries, the basics of complex manifolds, sheaves and cohomology theory, intersection theory on complex manifolds, Hermitian differential geometry, Kähler manifolds and Hodge theory. Chapter I contains the general theory of complex algebraic varieties. After the treatment of line bundles on complex manifolds and their Chern classes, the Kodaira vanishing theorem and the Lefschetz theorems, the question of algebraicity (Chow's theorem and Kodaira's embedding theorem) is discussed. Finally, the general theory is illustrated using Grassmann varieties and the Schubert calculus as guides. In Chapter II the first special class of algebraic varieties is considered—algebraic curves. This chapter includes most of the classical results on Riemann surfaces and algebraic curves, as for example the Hurwitz formula, Abel's theorem, linear systems on curves, canonical and hyperelliptic curves, Weierstrass points, the Brill-Noether problem and Plücker formulas. The connection between curves and abelian varieties via Jacobians is discussed, Andreotti's proof of the Torelli theorem is explained, and the projective embeddings of abelian varieties via theta functions are described. Chapter III is devoted to the development of further techniques necessary for the treatment of algebraic surfaces in Chapter IV. Here the theory of currents on analytic varieties, the theory of Chern classes for vector bundles, Lefschetz' fixed point formulas, Bott's residue formula and the general Hirzebruch-Riemann-Roch formula are all treated in full. The chapter ends with a section on spectral sequences and hypercohomology. In Chapter IV algebraic surfaces are considered. After the general theory, including intersection theory, blowing up and down and (bi-)rational maps the authors discuss numerous (classical) examples: quadric and cubic surfaces, the extensive theory of rational surfaces, ruled surfaces, elliptic surfaces, K3 surfaces and Enriques surfaces. The classification theory of surfaces due to Kodaira is also explained. The end of this chapter concerns singularities on surfaces and Noether's formula. Chapter V continues the development of basic techniques. It contains the following topics: theory and applications of residues, rudiments of commutative and homological algebra with applications, and global duality theory. To illustrate all the techniques developed so far in their mutual relations and to give some feeling for the treatment of higher-dimensional varieties, there is a final chapter entitled ``The quadric line complex''. This subject represents a part of algebraic geometry of long-standing attraction, and it is very useful and delightful to study. The chapter contains generalities on quadric hypersurfaces in Pn, lines and linear systems on quadrics, a discussion of the problem of five conics in the plane, the geometry of the Grassmannian G(2,4) of lines in projective 4-space, line complexes in P3, lines in quadric line complexes, associated abelian varieties and Kummer surfaces, and rationality considerations. This leads to the developing theory of higher-dimensional varieties via Fano varieties and intermediate Jacobians. After each chapter annotated references are given. Altogether this voluminous book is written in a beautiful, clear, and intuitive style. It is a very nice geometric counterpart to most ``algebraic'' books on algebraic geometry, and as well a good source as a reference book. The exposition is very inspiring for the reader and certainly should become a standard text.
Reviewer: Pfister, Gerhard [form MathSciNet]