The Arithmetic and Geometry of Algebraic Cycles

The NATO ASI/CRM Summer School at Banff offered a unique, full, and in-depth account of the topic, ranging from introductory courses by leading experts to discussions of the latest developments by all participants. The papers have been organized into three categories: cohomological methods; Chow groups and motives; and arithmetic methods. As a subfield of algebraic geometry, the theory of algebraic cycles has gone through various interactions with algebraic $K$-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology.These interactions have led to developments such as a description of Chow groups in terms of algebraic $K$-theory; the application of the Merkurjev-Suslin theorem to the arithmetic Abel-Jacobi mapping; progress on the celebrated conjectures of Hodge, and of Tate, which compute cycles class groups respectively in terms of Hodge theory or as the invariants of a Galois group action on etale cohomology; and, the conjectures of Bloch and Beilinson, which explain the zero or pole of the $L$-function of a variety and interpret the leading non-zero coefficient of its Taylor expansion at a critical point, in terms of arithmetic and geometric invariant of the variety and its cycle class groups. The immense recent progress in the theory of algebraic cycles is based on its many interactions with several other areas of mathematics. This conference was the first to focus on both arithmetic and geometric aspects of algebraic cycles. It brought together leading experts to speak from their various points of view. A unique opportunity was created to explore and view the depth and the breadth of the subject. This volume presents the intriguing results.

Table of Contents:
Cohomological Methods: Filtrations on the cohomology of abelian varieties by S. Abdulali Building mixed Hodge structures by D. Arapura The Atiyah-Chern character yields the semiregularity map as well as the infinitesimal Abdel-Jacobi map by R.-O. Buchweitz and H. Flenner Regulators and characteristic classes of flat bundles by J. Dupont, R. Hain, and S. Zucker Height pairings asymptotics and Bott-Chern forms by B. Harris and B. Wang Logarithmic Hodge structures and classifying spaces by K. Kato and S. Usui Chow Groups and Motives: Motives and algebraic de Rham cohomology by M. Asakura Hermitian vector bundles and characteristic classes by J. I. Burgos Gil The mixed motive of a projective variety by M. Hanamura Bloch's conjecture and the $K$-theory of projective surfaces by C. Pedrini From Jacobians to one-motives: Exposition of a conjecture of Deligne by N. Ramachandran Motives, algebraic cycles and Hodge theory by S. Saito Arithmetic methods: Picard-Fuchs uniformization: Modularity of the mirror map and mirror-moonshine by C. F. Doran Hilbert modular varieties in positive characteristic by E. Z. Goren On the Neron-Severi groups of some $K$3 surfaces by Y. Goto Torsion zero-cycles and the Abel-Jacobi map over the real numbers by J. van Hamel A remark on the Griffiths groups of certain product varieties by K. Kimura $p$-adic Abel-Jacobi maps and $p$-adic heights by J. Nekovar Crystalline fundamental groups and $p$-adic Hodge theory by A. Shiho Thompson series, and the mirror maps of pencils of $K$3 surfaces by H. Verrill and N. Yui.