Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. EUR 99.95/net; sFr. 166.00; £ 77.00; $ 129.00 (2004).

The first edition of this well-known and very popular standard text on compact complex surfaces was published in 1984, back then written by {\it W. Barth}, {\it C. Peters} and {\it A. Van de Ven} (1984;

Zbl 0718.14023). Apart from the modern treatises on complex surfaces by {\it I. R. Shafarevich}, {\it B. G. Averbukh}, {\it J. R. Vajnberg}, {\it A. B. Zizhchenko}, {\it Yu. I. Manin}, {\it B. G. Mojshezon}, {\it G. N. Tyurina}, {\it A. N. Tyurin} and {\it A. Beauville} [Astérisque 54, 1--172 (1978;

Zbl 0394.14014)], the first edition of the book under review offered the only up-to-date account of the subject in textbook form. Moreover, for almost twenty years it has been by far the most comprehensive textbook on complex surfaces from the modern point of view. The first edition contained eight main chapters on about 300 pages, concluding with the classification of K3 surfaces and Enriques surfaces.
The book under review is the second, substantially enlarged edition of this standard text, this time with K. Hulek as fourth co-author. In fact, in the two decades after the appearance of the first edition of the book, several crucial developments in the theory of complex surfaces have taken place, and the authors have taken the opportunity to update the original text by including some of those recent achievements.
The most important progress in the theory of complex surfaces has been made in regard of a better understanding of their differentiable structure (as real 4-manifolds), not at least in view of their appearance in mathematical physics. The new invariants discovered by Donaldson, on the one hand, and by Seiberg and Witten, on the other hand, stand for these spectacular recent developments. Other far-reaching achievements have been obtained by means of the study of nef-divisors on surfaces, parallel to progress made in the birational classification of higher-dimensional algebraic varieties, and also the Kähler structures on complex surfaces are now much better understood. Finally, I. Reider’s new approach to adjoint mappings, Bogomolov’s inequality for the Chern classes of rank-2 vector bundles on surfaces, and the mirror symmetry properties of K3 surfaces represent just as important new insights in the theory of complex surfaces.
Well, all these recent developments have been worked into the present new edition, in one way or another, and the text has grown to 436 pages, that is by more than forty percent. Apart from the correction of some minor irregularities in the first edition, the authors have left the well-proven original text basically intact. The enlargement of the material has been contrived by the addition of the new Chapter 9 entitled “Topological and Differentiable Structure of Surfaces”, including an introduction to Donaldson and Seiberg-Witten invariants (mainly by instructive examples), and by substantially enhancing Chapter 4 (“Some General Properties of Surfaces”). This chapter comes now with twelve (instead of eight) sections and includes the above-mentioned topics such as the nef cone, Bogomolov’s inequality, Reider’s method, and the existence of Kähler metrics on surfaces. There are also some other refining polishings and rearrangements in Chapter 5 (“Examples”) and Chapter 8 (“K3-Surfaces and Enriques Surfaces”). As to the contents of Chapter 8, three sections on special topics have been added, too, discussing the mirror symmetry phenomenon for projective K3-surfaces, special curves on K3-surfaces and an application to hyperbolic geometry, respectively.
Needless to say, the bibliography has been updated and tremendously enlarged, thereby reflecting the vast activity in the field during the past twenty years.
Now as before, the text is enriched by numerous instructive examples, but there are still no exercises for self-control, stimulus for further reading, or different outlooks.
All in all, the second, enlarged edition of this (meanwhile classic) textbook on complex surfaces has gained a good deal of topicality and disciplinary depth, while having maintained its high degree of systematic methodology, lucidity, rigor, and cultured style. No doubt, this book remains a must for everyone dealing with complex algebraic surfaces, be it a student, an active researcher in complex geometry, or a mathematically ambitioned (quantum) physicist.