Igor R. Shafarevich Basic Algebraic Geometry 2 Schemes and Complex Manifolds Third Edition Basic Algebraic Geometry 2 Igor R. Shafarevich Basic Algebraic Geometry 2 Schemes and Complex Manifolds Third Edition Igor R. Shafarevich Translator Algebra Section Miles Reid Steklov Mathematical Institute Mathematics Institute of the Russian Academy of Sciences University of Warwick Moscow, Russia Coventry, UK ISBN 978-3-642-38009-9 ISBN 978-3-642-38010-5 (eBook) DOI 10.1007/978-3-642-38010-5 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013945857 Mathematics Subject Classification (2010): 14-01 Translation of the 3rd Russian edition entitled “Osnovy algebraicheskoj geometrii”. MCCME, Moscow 2007, originally published in Russian in one volume © Springer-Verlag Berlin Heidelberg 1977, 1994, 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface to Books 2–3 Books 2–3 correspond to Chapters V–IX of the first edition. They study schemes and complex manifolds, two notions that generalise in different directions the varieties in projective space studied in Book 1. Introducing them leads also to new results in the theory of projective varieties. For example, it is within the framework of the theory of schemes and abstract varieties that we find the natural proof of the adjunc- tion formula for the genus of a curve, which we have already stated and applied in Section 2.3, Chapter 4. The theory of complex analytic manifolds leads to the study of the topology of projective varieties over the field of complex numbers. For some questions it is only here that the natural and historical logic of the subject can be re- asserted; for example, differential forms were constructed in order to be integrated, a process which only makes sense for varieties over the (real or) complex fields. Changes from the First Edition As in the Book 1, there are a number of additions to the text, of which the following two are the most important. The first of these is a discussion of the notion of the algebraic variety classifying algebraic or geometric objects of some type. As an example we work out the theory of the Hilbert polynomial and the Hilbert scheme. I am very grateful to V.I. Danilov for a series of recommendations on this subject. In particular the proof of Theorem 6.7 is due to him. The second addition is the definition and basic properties of a Kähler metric, and a description (without proof) of Hodge’s theorem. Prerequisites Varieties in projective space will provide us with the main supply of examples, and the theoretical apparatus of Book 1 will be used, but by no means all of it. Differ- ent sections use different parts, and there is no point in giving exact indications. References to the Appendix are to the Algebraic Appendix at the end of Book 1. V VI Preface to Books 2–3 Prerequisites for the reader of Books 2–3 are as follows: for Book 2, the same as for Book 1; for Book 3, the definition of differentiable manifold, the basic theory of analytic functions of a complex variable, and a knowledge of homology, cohomol- ogy and differential forms (knowledge of the proofs is not essential); for Chapter 9, familiarity with the notion of fundamental group and the universal cover. References for these topics are given in the text. Recommendations for Further Reading For the reader wishing to go further in the study of algebraic geometry, we can recommend the following references. For the cohomology of algebraic coherent sheaves and their applications: see Hartshorne [37]. An elementary proof of the Riemann–Roch theorem for curves is given in W. Ful- ton, Algebraic curves. An introduction to algebraic geometry, W.A. Benjamin, Inc., New York–Amsterdam, 1969. This book is available as a free download from http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf. For the general case of Riemann–Roch, see A. Borel and J.-P. Serre, Le théorème de Riemann–Roch, Bull. Soc. Math. France 86 (1958) 97–136, Yu.I. Manin, Lectures on the K-functor in algebraic geometry, Uspehi Mat. Nauk 24:5 (149) (1969) 3–86, English translation: Russian Math. Surveys 24:5 (1969) 1–89, W. Fulton and S. Lang, Riemann–Roch algebra, Grundlehren der mathematis- chen Wissenschaften 277, Springer-Verlag, New York, 1985. Moscow, Russia I.R. Shafarevich Contents Book 2: Schemes and Varieties 5 Schemes .................................. 3 1 The Spec of a Ring .......................... 5 1.1 Definition of Spec A ..................... 5 1.2 Properties of Points of Spec A ................ 7 1.3 The Zariski Topology of Spec A ............... 9 1.4 Irreducibility, Dimension .................. 11 1.5 Exercises to Section 1 .................... 14 2 Sheaves . ............................. 15 2.1 Presheaves.......................... 15 2.2 The Structure Presheaf . ................... 17 2.3 Sheaves ........................... 19 2.4 Stalks of a Sheaf ....................... 23 2.5 Exercises to Section 2 .................... 24 3 Schemes . ............................. 25 3.1 Definition of a Scheme ................... 25 3.2 Glueing Schemes . ................... 30 3.3 Closed Subschemes . ................... 32 3.4 Reduced Schemes and Nilpotents .............. 35 3.5 Finiteness Conditions . ................... 36 3.6 Exercises to Section 3 .................... 38 4 Products of Schemes ......................... 40 4.1 Definition of Product . ................... 40 4.2 Group Schemes ....................... 42 4.3 Separatedness ........................ 43 4.4 Exercises to Section 4 .................... 46 6 Varieties .................................. 49 1 DefinitionsandExamples...................... 49 1.1 Definitions.......................... 49 VII VIII Contents 1.2 Vector Bundles ........................ 53 1.3 Vector Bundles and Sheaves ................. 56 1.4 Divisors and Line Bundles .................. 63 1.5 Exercises to Section 1 .................... 67 2 AbstractandQuasiprojectiveVarieties............... 68 2.1 Chow’sLemma....................... 68 2.2 BlowupAlongaSubvariety................. 70 2.3 Example of Non-quasiprojective Variety . ......... 74 2.4 CriterionsforProjectivity.................. 79 2.5 Exercises to Section 2 .................... 81 3 Coherent Sheaves .......................... 81 3.1 Sheaves of OX-Modules ................... 81 3.2 Coherent Sheaves . ................... 85 3.3 Dévissage of Coherent Sheaves ............... 88 3.4 The Finiteness Theorem ................... 92 3.5 Exercises to Section 3 .................... 93 4 Classification of Geometric Objects and Universal Schemes .... 94 4.1 Schemes and Functors . ................... 94 4.2 The Hilbert Polynomial ...................100 4.3 Flat Families .........................103 4.4 The Hilbert Scheme . ...................107 4.5 Exercises to Section 4 ....................110 Book 3: Complex Algebraic Varieties and Complex Manifolds 7 The Topology of Algebraic Varieties ..................115 1 The Complex Topology .......................115 1.1 Definitions..........................115 1.2 Algebraic Varieties as Differentiable Manifolds; Orientation..........................117 1.3 Homology of Nonsingular Projective Varieties .......118 1.4 Exercises to Section 1 ....................121 2 Connectedness ............................121 2.1 PreliminaryLemmas.....................121 2.2 The First Proof of the Main Theorem . .........122 2.3 The Second Proof . ...................124 2.4 AnalyticLemmas......................126 2.5 Connectedness of Fibres ...................127 2.6 Exercises to Section 2 ....................128 3 The Topology of Algebraic Curves .................129 3.1 Local Structure of Morphisms ................129 3.2 Triangulation of Curves ...................131 3.3 Topological Classification of Curves . .........133 3.4 Combinatorial