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Zbl 0797.14002
Shafarevich, Igor R.
Basic algebraic geometry. 2: Schemes amd complex manifolds. Transl. from the Russian by Miles Reid. 2nd, rev. and exp. ed. (English)
[B] Berlin: Springer-Verlag. xv, 269 p. DM 68.00; öS 530.40; sFr. 68.00 (1994). ISBN 3-540-57554-5/pbk

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.\par 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.\par 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.
[W.Kleinert (Berlin)]

MSC 2000:: *14Axx Foundations of algebraic geometry
14C30 Transcendental methods
32Q99 Complex manifolds
14-02 Research monographs (algebraic geometry)
14-01 Textbooks (algebraic geometry)
14C05 Parametrization

Keywords: classifying spaces; moduli problems; Hilbert polynomial; Hilbert scheme; complex Kähler manifolds; Hodge theory

Citations: Zbl 0797.14001; Zbl 0258.14001

Cited in: Zbl pre06176095 Zbl 0878.14034 Zbl 0820.14022 Zbl 0797.14001 Zbl 1082.14501

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