Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds (Fields Institute Communications Book 67)

In recent years, research in K3 surfaces and Calabi–Yau varieties has seen spectacular progress from both arithmetic and geometric points of view, which in turn continues to have a huge influence and impact in theoretical physics—in particular, in string theory. The workshop on Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds, held at the Fields Institute (August 16-25, 2011), aimed to give a state-of-the-art survey of these new developments. This proceedings volume includes a representative sampling of the broad range of topics covered by the workshop. While the subjects range from arithmetic geometry through algebraic geometry and differential geometry to mathematical physics, the papers are naturally related by the common theme of Calabi–Yau varieties. With the big variety of branches of mathematics and mathematical physics touched upon, this area reveals many deep connections between subjects previously considered unrelated. Unlike most other conferences, the 2011 Calabi–Yau workshop started with 3 days of introductory lectures. A selection of 4 of these lectures is included in this volume. These lectures can be used as a starting point for the graduate students and other junior researchers, or as a guide to the subject. Front Matter....Pages i-xxvi Front Matter....Pages 1-1 K3 and Enriques Surfaces....Pages 3-28 Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi–Yau Varieties....Pages 29-69 Two Lectures on the Arithmetic of K3 Surfaces....Pages 71-99 Modularity of Calabi–Yau Varieties: 2011 and Beyond....Pages 101-139 Front Matter....Pages 141-141 Explicit Algebraic Coverings of a Pointed Torus....Pages 143-152 Elliptic Fibrations on the Modular Surface Associated to Γ 1 (8)....Pages 153-199 Universal Kummer Families Over Shimura Curves....Pages 201-265 Numerical Trivial Automorphisms of Enriques Surfaces in Arbitrary Characteristic....Pages 267-283 Picard–Fuchs Equations of Special One-Parameter Families of Invertible Polynomials....Pages 285-310 A Structure Theorem for Fibrations on Delsarte Surfaces....Pages 311-332 Fourier–Mukai Partners and Polarised $$\mathop{\mathrm{K3}}\nolimits$$ Surfaces....Pages 333-365 On a Family of K3 Surfaces with $$\mathcal{S}_{4}$$ Symmetry....Pages 367-386 K 1 ind of Elliptically Fibered K 3 Surfaces: A Tale of Two Cycles....Pages 387-409 A Note About Special Cycles on Moduli Spaces of K3 Surfaces....Pages 411-427 Enriques Surfaces of Hutchinson–Göpel Type and Mathieu Automorphisms....Pages 429-454 Quartic K3 Surfaces and Cremona Transformations....Pages 455-460 Invariants of Regular Models of the Product of Two Elliptic Curves at a Place of Multiplicative Reduction....Pages 461-487 Front Matter....Pages 489-489 Dynamics of Special Points on Intermediate Jacobians....Pages 491-498 Calabi–Yau Conifold Expansions....Pages 499-515 Quadratic Twists of Rigid Calabi–Yau Threefolds Over Q ....Pages 517-533 Front Matter....Pages 489-489 Counting Sheaves on Calabi–Yau and Abelian Threefolds....Pages 535-548 The Segre Cubic and Borcherds Products....Pages 549-565 Quasi-modular Forms Attached to Hodge Structures....Pages 567-587 The Zero Locus of the Infinitesimal Invariant....Pages 589-602 In recent years, research in K3 surfaces and Calabi–Yau varieties has seen spectacular progress from both the arithmetic and geometric points of view, which in turn continues to have a huge influence and impact in theoretical physics—in particular, in string theory. The workshop on Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds, held at the Fields Institute (August 16–25, 2011), aimed to give a state-of-the-art survey of these new developments. This proceedings volume includes a representative sampling of the broad range of topics covered by the workshop. While the subjects range from arithmetic geometry through algebraic geometry and differential geometry to mathematical physics, the papers are naturally related by the common theme of Calabi–Yau varieties. With the large variety of branches of mathematics and mathematical physics touched upon, this area reveals many deep connections between subjects previously considered unrelated. Unlike most other conferences, the 2011 Calabi–Yau workshop started with three days of introductory lectures. A selection of four of these lectures is included in this volume. These lectures can be used as a starting point for graduate students and other junior researchers, or as a guide to the subject. This book surveys the very active field of Calabi-Yau varieties from a geometric and arithmetic perspective. It includes four introductory lectures that can be used by graduate students and other researchers as a guide to the field and contains a varied selection of topics from pure arithmetic questions to geometric questions to Hodge theory Arising From A 2011 Workshop At The Fields Institute, This Book Reviews Arithmetic And Geometry Of K3 Surfaces And Calabi-yau Threefolds. Offers Lectures And Papers On Arithmetic And Algebraic Geometry, Differential Geometry, Mathematical Physics And More. Radu Laza, Matthias Schütt, Noriko Yui, Editors. Includes Bibliographical References.