**Fields Institute Communications**

Volume: 38;
2003;
367 pp;
Hardcover

MSC: Primary 14;

Print ISBN: 978-0-8218-3355-1

Product Code: FIC/38

List Price: $129.00

Individual Member Price: $103.20

**Electronic ISBN: 978-1-4704-3072-6
Product Code: FIC/38.E**

List Price: $129.00

Individual Member Price: $103.20

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# Calabi-Yau Varieties and Mirror Symmetry

Share this page *Edited by *
*Noriko Yui; James D. Lewis*

A co-publication of the AMS and Fields Institute

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.

Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Table of Contents

# Table of Contents

## Calabi-Yau Varieties and Mirror Symmetry

- Cover Cover11
- Title page i2
- Contents iii4
- Schedule of workshops v6
- List of participants vii8
- Acknowledgments ix10
- Introduction xi12
- Geometric methods 118
- Mixed toric residues and Calabi-Yau complete intersections 320
- Crepant resolutions of ℂⁿ/𝔸₁(𝕟) and flops of 𝕟-folders for 𝕟=4,5 2744
- Picard-Fuchs equations, integrable systems and higher algebraic K-theory 4360
- Counting BPS states via holomorphic anomaly equations 5774
- Regulators of Chow cycles on Calabi-Yau varieties 87104
- Arithmetic methods 119136
- Calabi-Yau manifolds over finite fields, II 121138
- Modularity of rigid Calabi-Yau threefolds over ℚ 159176
- 𝐾3 surfaces with symplectic group actions 167184
- Birational smooth minimal models have equal Hodge numbers in all dimensions 183200
- The 𝑛th root of the mirror map 195212
- On a Shioda-Inose structure of a family of K3 surfaces 201218
- Black hole attractor varieties and complex multiplication 209226
- Hypergeometric families of Calabi-Yau manifolds 223240
- Aspects of conformal field theory from Calabi-Yau arithmetic 233250
- Ordinary Calabi-Yau-3 crystals 255272
- The ordinary limit for varieties over ℤ[𝕩₁,…,𝕩ᵣ] 273290
- Update on the modularity of Calabi-Yau varieties with appendix by Helena Verrill 307324
- Problems 363380
- Back Cover Back Cover1385

#### Readership

Graduate students and research mathematicians interested in mirror symmetry and string theory.