THE HODGE THEORY OF PROJECTIVE MAN I FOLDS TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk THE HODGE THEORY OF PROJ ECTIVE MAN I FOLDS MARK ANDREA DE CATALDO Stony Brook University, USA Imperial College Press Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. THE HODGE THOERY OF PROJECTIVE MANIFOLDS Copyright © 2007 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13978-1-86094-800-8 ISBN-101-86094-800-6 Printed in Singapore. EH - The Hodge Theory.pmd 1 2/26/2007, 4:04 PM January29,2007 22:22 WorldScientificBook-9inx6in test Alla memoria di Meeyoung v January29,2007 22:22 WorldScientificBook-9inx6in test TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk January29,2007 22:22 WorldScientificBook-9inx6in test Preface After the mountain, another mountain (traditional Korean saying) AsmyfriendandcolleagueLucaMigliorinioncewrotetome,thetopol- ogy of algebraic varieties is a mystery and a miracle. These lectures are an attempt to introduce the reader to the Hodge theory of algebraic varieties. The geometric implications of Hodge theory for a compact oriented manifold become progressively richer and more beautiful as one specializes fromRiemannian,tocomplex,toK¨ahlerandfinallytoprojectivemanifolds. The structure of these lectures tries to reflect this fact. Ideliveredeightone-hourlecturesattheJuly22–July27,2003Summer School on Hodge Theory at the Byeonsan Peninsula in South Korea. The presenttextisasomewhatexpandedanddetailedversionofthoselectures. IwouldliketothankProfessorJongHaeKeumandDr. ByungheupJun for organizing the event, for editing a privately circulated version of these notes and for kindly agreeing to their publication. IwouldliketothankProfessorJun-MukHwangandProfessorYongnam Lee for supporting the Summer School. I would like to thank all of those who have attended the lectures for the warm atmosphere I have found that has made my stay a wonderful vii January29,2007 22:22 WorldScientificBook-9inx6in test viii Lectures on the Hodge Theory of Projective Manifolds experience. In particular, I would like to thank Professor Dong-Kwan Shin for showing me with great humor some aspects of the Korean culture. Beyond the choice of topics, exercises and exposition style, nothing in these written-up version of the lectures is original. The reader is assumed to have some familiarity with smooth and complex manifolds. These lec- tures are not self-contained and at times a remark or an exercise require knowledgeofnotionsandfactswhicharenotcoveredhere. Iseenoharmin that, but rather as an encouragement to the reader to explore the subjects involved. Thelocationoftheexercisesinthelecturesisasuggestiontothe reader to solve them before proceeding to the theoretical facts that follow. The table of contents should be self-explanatory. The only exception is §8 where I discuss, in a simple example, a technique for studying the class mapforhomologyclassesonthefibersofamapandoneforapproximating a certain kind of primitive vectors. These techniques have been introduced in [de Cataldo and Migliorini, 2002] and [de Cataldo and Migliorini, 2005]. I would like to thank Fiammetta Battaglia, N. Hao, Gabriele LaNave, Jungool Lee and Luca Migliorini for suggesting improvements and correct- ing some mistakes. The remaining mistakes and the shortcomings in the exposition are mine. During the preparation of these lecture notes, I have been supported by the following grants: N.S.F. Grant DMS 0202321, N.S.F. Grant DMS 0501020 and N.S.A. Grant MDA904-02-1-0100. The final version of the book was completed at I.A.S. (Princeton), under grant DMS 0111298. Mark Andrea Antonio de Cataldo February27,2007 14:46 WorldScienti(cid:12)cBook-9inx6in test Contents Preface vii 1. Calculus on smooth manifolds 1 1.1 The Euclidean structure on the exterior algebra . . . . . . 1 1.2 The star isomorphism on (cid:3)(V) . . . . . . . . . . . . . . . 2 1.3 The tangent and cotangent bundles of a smooth manifold 4 1.4 The de Rham cohomologygroups . . . . . . . . . . . . . . 6 1.5 Riemannian metrics . . . . . . . . . . . . . . . . . . . . . 11 1.6 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . 12 1.7 Orientation and integration . . . . . . . . . . . . . . . . . 12 2. The Hodge theory of a smooth, oriented, compact Riemannian manifold 19 2.1 The adjoint of d: d? . . . . . . . . . . . . . . . . . . . . . 19 2.2 TheLaplace-Beltramioperatorof anoriented Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Harmonic forms and the Hodge Isomorphism Theorem . . 22 3. Complex manifolds 27 3.1 Conjugations . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Tangent bundles on a complex manifold . . . . . . . . . . 28 3.3 Cotangent bundles on complex manifolds . . . . . . . . . 31 3.4 The standard orientation of a complex manifold . . . . . . 33 3.5 The quasi complex structure . . . . . . . . . . . . . . . . 34 3.6 Complex-valued forms . . . . . . . . . . . . . . . . . . . . 37 3.7 Dolbeault and Bott-Chern cohomology . . . . . . . . . . . 40 ix