Basic Algebraic Geometry 2: Schemes and Complex Manifolds: Third Edition

Main subject categories: • Algebraic geometry • Schemes • Complex manifoldsMathematics Subject Classification (2010): • 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometryShafarevich'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. 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. 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, 'F̀or 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, Shafarevichs book is a must.'' The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann--Roch theorem for curves, including a proof from first principles. Shafarevich's book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field 1. Varieties In Projective Space -- 2. Schemes And Complex Manifolds. Igor R. Shafarevich. Translation Of The 3rd Russian Edition Entitled Osnovy Algebraicheskoj Geometrii ... Originally Published In Russian In One Volume--title Page Verso. Translator: Miles A. Reid. Includes Bibliographical References And Indexes. Front Matter....Pages I-XIV Front Matter....Pages 1-1 Schemes....Pages 3-47 Varieties....Pages 49-111 Front Matter....Pages 113-113 The Topology of Algebraic Varieties....Pages 115-148 Complex Manifolds....Pages 149-199 Uniformisation....Pages 201-228 Back Matter....Pages 229-262 Basic Algebraic Geometry II is a revised edition of Shafarevich's well-known introductory book on algebraic varieties and complex manifolds. It can be read independently of Volume I and is suitable for graduate students in mathematics and theoretical physics.