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Complex algebraic surfaces. 2nd ed. (English) Zbl 0849.14014
London Mathematical Society Student Texts. 34. Cambridge: Cambridge Univ. Press. ix, 132 p., £37.50; $ 59.95/hbk (1996).
The French original of the author’s introductory text on the Enriques classification of complex algebraic surfaces appeared in 1978 (cf. Zbl 0394.14014). Grown out of a graduate course on this classical, nevertheless still very active area of research in geometry, the book aimed at giving a comparatively elementary and concise account on the geometry of complex algebraic surfaces and their birational classification theory. Using the modern methods of algebraic geometry, in particular sheaves and their cohomology, the author succeeded in presenting F. Enriques’s program of classifying surfaces in a lucid, systematic and methodically updated manner. Being perfectly suited for students and non-experts interested in the subject, this textbook became very soon a widely used standard book on complex surfaces, and that for both students and teachers at universities.
The first English edition, that is a faithful translation of the French original, appeared in 1983 (cf. Zbl 0512.14020), making this outstanding text available (and accessible) for a much larger group of users. – The present second (English) edition is a reprint of the first (English) edition, enlarged by two more short appendices, and by some related recent references in the bibliography. The new appendix B, entitled “Complex surfaces”, gives an outlook to the Kodaira classification of complex analytic surfaces, and may serve as an invitation to study the more recent standard text by W. Barth, C. Peters and A. Van de Ven: “Compact complex surfaces” (1984; Zbl 0718.14023). Appendix C, the concluding section of this new edition, points out some different approaches to the classification of surfaces and a few recent developments. As the title of this appendix says, the reader is here provided with hints for further reading.
Undoubtedly, this second (English) edition of Beauville’s well-proved text on complex algebraic surfaces will maintain its great value within the existing literature on the subject.

14J25 Special surfaces
32J15 Compact complex surfaces
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14J10 Families, moduli, classification: algebraic theory
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces