*(English)*Zbl 0836.14001

The development of algebraic geometry in the 20th century is characterized by several radical changes in style and language. The process that led to algebraic geometry at its present stage started in the later 1940s, when sheaf theory and cohomology theories emerged as a basic toolkit in various branches of mathematics, and it culminated in the 1960s with A. Grothendieck’s new foundation of algebraic geometry via the abstract-algebraic theory of schemes.

The first textbooks on algebraic geometry, which reflected these new concepts and methods in a systematic and (more or less) comprehensive way, appeared in the 1970s. After *I. R. Shafarevich*’s “Basic algebraic geometry” (in Russian 1972; Zbl 0258.14001; English edition 1974), *D. Mumford*’s “Algebraic geometry. I” (1976; Zbl 0356.14002) and *R. Hartshorne*’s “Algebraic geometry” (1977; Zbl 0367.14001) had appeared, the authors published their “Principles of algebraic geometry” first in 1978 (cf. Zbl 0408.14001). Their treatise differed from the others in many regards and offered many particular features.

First of all, the book under review is a text on complex algebraic geometry. It does not stress the most general abstract-algebraic approach via schemes and their categorical sheaf theory. Instead, it focuses on the transcendental aspects of complex projective varieties, that is on their underlying Kähler geometry, Hodge theory and Kodaira-Lefschetz theory. Secondly, this transcendental approach is organically combined with the classical projective geometry of algebraic varieties, including various classical topics such as Grassmannians, enumerative formulae, varieties of lines, and others. – Finally, apart from the presentation of the wide spectrum of modern analytic and algebraic methods, together with their application to the study of complex projective manifolds, the book offers a particularly nice and comprehensive discussion of the classical theories of algebraic curves and surfaces, and that from both the modern and the classical viewpoint.

These features give the book under review its unique character among the (meanwhile) many existing textbooks on modern algebraic geometry. Now as before, it represents the by far most valuable and complete supplement to the more general and astract textbooks on algebraic geometry, with regard to the transcendental and classic-projective topics, and it contains a wealth of geometric-intuitive ideas, ingenious arguments and particular results that cannot be found in any other text For both researchers and students this book is still an inexhaustible and indispensible source.

The present edition is a reprint of the original text, without any changes. Being one of the really great classics of algebraic geometry, this text is timelessly outstanding and does not need any additional enlargement or any modifications.

##### MSC:

14-02 | Research monographs (algebraic geometry) |

14Jxx | Surfaces and higher-dimensional varieties |

32Jxx | Compact analytic spaces |

14Hxx | Algebraic curves |

14C30 | Transcendental methods, Hodge theory, Hodge conjecture |