Table Of Content

Selected Titles in This Series Volume 9 Shing-Tung Yau, Editor Minor Symmetry I 1998 8 Jürgen Jost, Wiifrid Kendail, Umberto Mosco, Michael Röckner, and Karl-Theodor Sturm New Directions in Dirichlet Forms 1998 7 D. A. Buell and J. T. Teitelbaum, Editors Computational Perspectives on Number Theory 1998 6 Harold Levine Partial Differential Equations 1997 5 Qi-keng Lu, Stephen S.-T. Yau, and Anatoly Libgober, Editors Singularities and Complex Geometry 1997 4 Vyjayanthi Chari and Ivan B. Penkov, Editors Modular Interfaces: Modular Lie Algebras, Quantum Groups, and Lie Superalgebras 1997 3 Xia-Xi Ding and Tai-Ping Liu, Editors Nonlinear Evolutionary Partial Differential Equations 1997 2.2 William H. Kazez, Editor Geometrie Topology 1997 2.1 William H. Kazez, Editor Geometrie Topology 1997 1 B. Greene and S.-T. Yau, Editors Mirror Symmetry II 1997 This page intentionally left blank Mirror Symmetry I This page intentionally left blank AMS/IP https://doi.org/10.1090/amsip/009 Studies in Advanced Mathematics Volume 9 Mirror Symmetry I Shing-Tung Yau, Editor American Mathematical Society · International Press Shing-Tung Yau, Managing Editor 2000 Mathematics Subject Classification. Primary 14-XX; Secondary 32-XX, 81-XX. Library of Congress Cataloging-in-Publication Data Mirror symmetry I / Shing-Tung Yau, editor. p. cm. — (AMS/IP studies in advanced mathematics, ISSN 1089-3288 ; v. 9) Includes bibliographical references. ISBN 0-8218-0665-3 (alk. paper) 1. Mirror symmetry. 2. Manifolds (Mathematics) I. Yau. Shing-Tung, 1949- IL Series. QC174.17.S9M562 1998 98-35520 516.3,62-dc21 CIP Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by Services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission®ams. org. Excluded from these provisions is material in articles for which the author holds Copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1998 by the American Mathematical Society and International Press. All rights reserved. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ Visit the International Press home page at URL: http://www.intlpress.com/ 10 9 8 7 6 5 4 3 2 06 05 04 03 02 01 Contents Preface ix Introduction xi An introduction to minor manifolds B. R. GREENE AND M, R. PLESSER 1 Á pair of Calabi-Yau manifolds as an exactly soluble superconformal theory PHILIP CANDELAS, XENIA C. DE LA OSSA, PAUL S. GREEN, AND LINDA PARKES 31 Topological mirrors and quantum rings CUMRUN VAFA 97 Mirror manifolds and topological field theory EDWARD WITTEN 121 Rational curves and Classification of algebraic varieties YUJIRO KAWAMATA 161 Rational curves on Calabi-Yau threefolds SHELDON KATZ 169 Picard-Fuchs equations and mirror maps for hypersurfaces DAVID R. MORRISON 185 Kahler classes on Calabi-Yau threefolds—An informal survey P. M. H. WILSON 201 Automorphic functions and special Kahler geometry S. FERRARA, C. KOUNNAS, D. LUST, AND F. ZWIRNER 215 Picard-Fuchs equations and flat holomorphic connections from Í = 2 supergravity S. FERRARA AND J. LOUIS 237 Á new geometry from superstring theory P. S. ASPINWALL AND C A . LÜTKEN 253 The geometry of Calabi-Yau orbifolds SHI-SHYR ROAN 279 Properties of superstring vacua from (topological) Landau-Ginzburg modeis AMIT GIVEON AND DIRK-JAN SMIT 287 viii CONTENTS An 51/(2, C) action on certain Jacobian rings and the mirror map TRISTAN HÜBSCH AND SHING-TUNG YAU 311 Á generalized construction of mirror manifolds PER BERGLUND AND TRISTAN HÜBSCH 327 New constructions of mirror manifolds: Probing moduli space far from Fermat points B. R. GREENE, M. R. PLESSER, AND S.-S. ROAN 347 Deformations of Calabi-Yau Kleinfolds Ziv RAN 391 Smoothing 3-folds with trivial canonical bündle and ordinary double points GANG TIAN 399 Modified Calabi-Yau manifolds with torsion MARTIN ROCEK 421 Calabi-Yau threefolds and complex multiplication ClPRIAN BORCEA 431 Preface to the New Edition This volume represents a new edition of the papers which were originally published m Essays on Mirror Manifolds. That volume has continued to be quite populär, and this new edition is supplemented by an additional volume, Mirror Symmetry 2, edited by B. Greene and myself. In this volume, we would like to address a number of errors which were found in the original: for example, the Workshop on Mirror Manifolds at the Mathematical Sciences Research Institute took place in 1991, not 1911. The paper by Geir Ellingsrud and Stein A. Stromme has been withdrawn at the authors' request. Meanwhile, P.M.H. Wilson's paper on "Kahler Classes on Calabi-Yau Threefolds- an Informal Survey"; Sheldon Katz's paper on "Rational Curves on Calabi-Yau Threefolds"; Ziv Ran's paper on "Deformations of Calabi-Yau Kleinfolds"; and David R. Morrison's paper on "Picard-Puchs Equations and Mirror Maps for Hypersurfaces" have all been corrected per new drafts from the authors. We are quite pleased with the continuing interest in each of the papers in this volume and look forward to the continuing development of this field in the new Century. S.-T. Yau Cambridge, MA