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Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.'' The second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the ``Shafarevich conjecture''. The style of  Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of  Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.

Geometry, algebraic. --- Mathematics. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Algebraic geometry. --- Physics. --- Algebraic Geometry. --- Theoretical, Mathematical and Computational Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Algebraic geometry --- Math --- Science --- Geometry, Algebraic. --- Mathematical physics. --- Physical mathematics --- Physics --- Geometria algebraica

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From the reviews: "This is one of the few mathematical books, the reviewer has read from cover to cover ... The main merit is that nearly on every page you will find some unexpected insights..." Zentralblatt für Mathematik und Ihre Grenzgebiete, 1991 "...which I read like a novel and undoubtedly will become a classic. ... A merit of the book under review is that it contains several important articles from journals which are not all so easily accessible. ... Furthermore, at the end of the book, there are some Notes by the author which are indispensable for the necessary historical background information. ... This valuable book should be on the shelf of every algebraist and algebraic geometer." Nieuw Archief voor Wiskunde, 1992 "... There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition ... which transports the reader effortlessly across the whole spectrum of algebra.... The challenge to Ezekiel, "Can these bones live?" is, all too often, the reaction of students when introduced to the bare bones of the concepts and constructs of modern algebra. Shafarevich's book - which reads as comfortably as an extended essay - breathes life into the skeleton and will be of interest to many classes of readers..." The Mathematical Gazette, 1991 "... According to the preface, the book is addressed to "students of mathematics in the first years of an undergraduate course, or theoretical physicists or mathematicians from outside algebra wanting to get an impression of the spirit of algebra and its place in mathematics." I think that this promise is fully justified. The beginner, the experts and also the interested scientist who had contact with algebraic notions - all will read this exceptional book with great pleasure and benefit." Zeitschrift für Kristallographie, 1991.

Algebra. --- Mathematics. --- Math --- Science --- Mathematics --- Mathematical analysis --- Algebra --- Topological Groups. --- K-theory. --- Topological Groups, Lie Groups. --- K-Theory. --- Algebraic topology --- Homology theory --- Groups, Topological --- Continuous groups --- Topological groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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From the reviews: "This is one of the few mathematical books, the reviewer has read from cover to cover ... The main merit is that nearly on every page you will find some unexpected insights..." Zentralblatt für Mathematik und Ihre Grenzgebiete, 1991 "...which I read like a novel and undoubtedly will become a classic. ... A merit of the book under review is that it contains several important articles from journals which are not all so easily accessible. ... Furthermore, at the end of the book, there are some Notes by the author which are indispensable for the necessary historical background information. ... This valuable book should be on the shelf of every algebraist and algebraic geometer." Nieuw Archief voor Wiskunde, 1992 "... There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition ... which transports the reader effortlessly across the whole spectrum of algebra.... The challenge to Ezekiel, "Can these bones live?" is, all too often, the reaction of students when introduced to the bare bones of the concepts and constructs of modern algebra. Shafarevich's book - which reads as comfortably as an extended essay - breathes life into the skeleton and will be of interest to many classes of readers..." The Mathematical Gazette, 1991 "... According to the preface, the book is addressed to "students of mathematics in the first years of an undergraduate course, or theoretical physicists or mathematicians from outside algebra wanting to get an impression of the spirit of algebra and its place in mathematics." I think that this promise is fully justified. The beginner, the experts and also the interested scientist who had contact with algebraic notions - all will read this exceptional book with great pleasure and benefit." Zeitschrift für Kristallographie, 1991.

#WBIB:dd.Lic.L.De Busschere --- 512.5 --- General algebra --- 512.5 General algebra --- Topological groups. --- Lie groups. --- K-theory. --- Topological Groups, Lie Groups. --- K-Theory.