Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

Christian Rohde

ISBN 10: 3642006388 / ISBN 13: 9783642006388

Published by Springer Berlin Heidelberg, Springer Berlin Heidelberg Apr 2009, 2009

About this Item

Neuware -Calabi-Yau manifolds have been an object of extensive research during the last two decades. One of the reasons is the importance of Calabi-Yau 3-manifolds in modern physics - notably string theory. An interesting class of Calabi-Yau manifolds is given by those with complex multiplication (CM). Calabi-Yau manifolds with CM are also of interest in theoretical physics, e. g. in connection with mirror symmetry and black hole attractors. It is the main aim of this book to construct families of Calabi-Yau 3-manifolds with dense sets of bers with complex multiplication. Most - amples in this book are constructed using families of curves with dense sets of bers with CM. The contents of this book can roughly be divided into two parts. The rst six chapters deal with families of curves with dense sets of CM bers and introduce the necessary theoretical background. This includes among other things several aspects of Hodge theory and Shimura varieties. Using the rst part, families of Calabi-Yau 3-manifolds with dense sets of bers withCM are constructed in the remaining ve chapters. In the appendix one nds examples of Calabi-Yau 3-manifolds with complex mul- plication which are not necessarily bers of a family with a dense set ofCM bers. The author hopes to have succeeded in writing a readable book that can also be used by non-specialists.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 244 pp. Englisch.

Seller Inventory # 9783642006388

Bibliographic Details

• Title: Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication
• Publisher: Springer Berlin Heidelberg, Springer Berlin Heidelberg Apr 2009
• Publication Date: 2009
• Language: English
• ISBN 10: 3642006388
• ISBN 13: 9783642006388
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 235W x 155H x 14D millimeters 235 millimeters width by 155 millimeters height by 14 millimeters depth
• Weight: 376 grams

About this title

Synopsis

This book constructs families of Calabi-Yau 3-manifolds with dense sets of complex multiplication fibers, which are determined by combining the method of E. Viehweg and K. Zuo with the construction method of the Borcea-Voisin mirror family.

From the Back Cover

The main goal of this book is the construction of families of Calabi-Yau 3-manifolds with dense sets of complex multiplication fibers. The new families are determined by combining and generalizing two methods.

Firstly, the method of E. Viehweg and K. Zuo, who have constructed a deformation of the Fermat quintic with a dense set of CM fibers by a tower of cyclic coverings. Using this method, new families of K3 surfaces with dense sets of CM fibers and involutions are obtained.

Secondly, the construction method of the Borcea-Voisin mirror family, which in the case of the author's examples yields families of Calabi-Yau 3-manifolds with dense sets of CM fibers, is also utilized. Moreover fibers with complex multiplication of these new families are also determined.

This book was written for young mathematicians, physicists and also for experts who are interested in complex multiplication and varieties with complex multiplication. The reader is introduced to generic Mumford-Tate groups and Shimura data, which are among the main tools used here. The generic Mumford-Tate groups of families of cyclic covers of the projective line are computed for a broad range of examples.