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**Basic algebraic geometry. 2: Schemes amd complex manifolds. Transl. from the Russian by Miles Reid.
2nd, rev. and exp. ed.**
*(English)*
Zbl 0797.14002

Berlin: Springer-Verlag. xv, 269 p. (1994).

The second volume of the new edition of this textbook is an expanded version of chapters V–IX of the first edition (1972; Zbl 0258.14001). Accordingly, it is devoted to the foundations of the theory of algebraic schemes and the theory of complex algebraic varieties. As in the first volume, the author has enriched the original material by some additional and important topics, leaving the well-established disposition of the original version essentially intact.

The first major addition concerns those algebraic varieties that serve as classifying spaces for certain algebro-geometric objects. In other words, the viewpoint of families and moduli problems is now brought in. In particular, the theory of the Hilbert polynomial and the concept of the Hilbert scheme have been worked into chapter VI (“Varieties”). The other remarkable insertion deals with the basic theory of complex Kähler manifolds and contains, among other things, a survey on Hodge theory. Generally, the method of vector bundles and the basic techniques of Hermitean differential geometry come more decisively into play. In this way, the interplay between algebraic geometry and complex analysis is particularly emphasized, much more than in the first edition, and this fascinating correlation is precisely what the author intentionally wanted to stress.

Altogether, as for this second volume, one can only respectfully repeat what has been said about the first part of the book (see the preceding review): a great textbook, written by one of leading algebraic geometers and teachers himself, has been reworked and updated. As a result the author’s standard textbook on algebraic geometry has become even more important and valuable. Students, teachers, and active researchers using methods of algebraic and complex-analytic geometry in different areas of mathematics and theoretical physics should be grateful to the author for his renewed service to the mathematical community.

The first major addition concerns those algebraic varieties that serve as classifying spaces for certain algebro-geometric objects. In other words, the viewpoint of families and moduli problems is now brought in. In particular, the theory of the Hilbert polynomial and the concept of the Hilbert scheme have been worked into chapter VI (“Varieties”). The other remarkable insertion deals with the basic theory of complex Kähler manifolds and contains, among other things, a survey on Hodge theory. Generally, the method of vector bundles and the basic techniques of Hermitean differential geometry come more decisively into play. In this way, the interplay between algebraic geometry and complex analysis is particularly emphasized, much more than in the first edition, and this fascinating correlation is precisely what the author intentionally wanted to stress.

Altogether, as for this second volume, one can only respectfully repeat what has been said about the first part of the book (see the preceding review): a great textbook, written by one of leading algebraic geometers and teachers himself, has been reworked and updated. As a result the author’s standard textbook on algebraic geometry has become even more important and valuable. Students, teachers, and active researchers using methods of algebraic and complex-analytic geometry in different areas of mathematics and theoretical physics should be grateful to the author for his renewed service to the mathematical community.

Reviewer: W.Kleinert (Berlin)

### MSC:

14Axx | Foundations of algebraic geometry |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

32Q99 | Complex manifolds |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14C05 | Parametrization (Chow and Hilbert schemes) |