Calabi-Yau Varieties and Mirror Symmetry

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Overview

The idea of mirror symmetry originated in physics, but in recent years, the field of mirror symmetry has exploded onto the mathematical scene. It has inspired many new developments in algebraic and arithmetic geometry, toric geometry, the theory of Riemann surfaces, and infinite-dimensional Lie algebras among others. The developments in physics stimulated the interest of mathematicians in Calabi-Yau varieties. This led to the realization that the time is ripe for mathematicians, armed with many concrete examples and alerted by the mirror symmetry phenomenon, to focus on Calabi-Yau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for Calabi-Yau threefolds defined over the rationals, the Bloch-Beilinson conjectures, regulator maps of higher algebraic cycles, Picard-Fuchs differential equations, GKZ hypergeometric systems, and others. The articles in this volume report on current developments. The papers are divided roughly into two categories: geometric methods and arithmetic methods. One of the significant outcomes of the workshop is that we are finally beginning to understand the mirror symmetry phenomenon from the arithmetic point of view, namely, in terms of zeta-functions and L-series of mirror pairs of Calabi-Yau threefolds. The book is suitable for researchers interested in mirror symmetry and string theory.

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Product Details

  • ISBN-13: 9780821833551
  • Publisher: American Mathematical Society
  • Publication date: 9/9/2003
  • Series: Fields Institute Communications Series , #38
  • Pages: 367
  • Product dimensions: 72.50 (w) x 10.25 (h) x 1.00 (d)

Table of Contents

Schedule of Workshops
List of Participants
Acknowledgments
Introduction
Mixed Toric Residues and Calabi-Yau Complete Intersections 3
Crepant Resolutions of C[superscript n]/A[subscript 1](n) and Flops of n-folds for n = 4,5 27
Picard-Fuchs Equations, Integrable Systems and Higher Algebraic K-theory 43
Counting BPS States via Holomorphic Anomaly Equations 57
Regulators of Chow Cycles on Calabi-Yau Varieties 87
Calabi-Yau Manifolds over Finite Fields, II 121
Modularity of Rigid Calabi-Yau Threefolds over Q 159
K3 Surfaces with Symplectic Group Actions 167
Birational Smooth Minimal Models have Equal Hodge Numbers in all Dimensions 183
The nth Root of the Mirror Map 195
On a Shioda-Inose Structure of a Family of K3 Surfaces 201
Black Hole Attractor Varieties and Complex Multiplication 209
Hypergeometric Families of Calabi-Yau Manifolds 223
Aspects of Conformal Field Theory from Calabi-Yau Arithmetic 233
Ordinary Calabi-Yau-3 Crystals 255
The Ordinary Limit for Varieties over Z[x[subscript 1],...,x[subscript r]] 273
Update on the Modularity of Calabi-Yau Varieties 307
Problems 363
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