Table Of ContentSelected 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
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Mirror
Symmetry I
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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
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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
