Algebraic Geometry over the Complex Numbers
This textbook is a strong addition to existing introductory literature on algebraic geometry. The author’s treatment combines the study of algebraic geometry with differential and complex geometry and unifies these subjects using sheaf-theoretic ideas. It
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Universitext Series Editors: Sheldon Axler San Francisco State University Vincenzo Capasso Università degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus J. MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah CNRS, École Polytechnique Endre Süli University of Oxford Wojbor A. Woyczynski Case Western Reserve University
Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. For further volumes: http://www.springer.com/series/223
Donu Arapura
Algebraic Geometry over the Complex Numbers
Donu Arapura Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907 USA [email protected]
ISSN 0172-5939 e-ISSN 2191-6675 ISBN 978-1-4614-1808-5 e-ISBN 978-1-4614-1809-2 DOI 10.1007/978-1-4614-1809-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2012930383 Mathematics Subject Classification (2010): 14-XX, 14C30 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my parents, who taught me that knowledge is something to be valued
Preface
Algebraic geometry is the geometric study of sets of solutions to polynomial equations over a field (or ring). These objects, called algebraic varieties (or schemes or . . . ), can be studied using tools from commutative and homological algebra. When the field is the field of complex numbers, these methods can be supplemented with transcendental ones, that is, by methods from complex analysis, differential geometry, and topology. Much of the beauty of the subject stems from the rich interplay of these various techniques and viewpoints. Unfortunately, this also makes it a hard subject to learn.
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