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The Arithmetic and Geometry of Algebraic Cycles
Edited by: B. Brent Gordon, University of Oklahoma, Norman, OK, James D. Lewis, University of Alberta, Edmonton, AB, Canada, Stefan Müller-Stach, Universität Essen, Germany, Shuji Saito, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Japan, and Noriko Yui, Queen's University, Kingston, ON, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Proceedings & Lecture Notes
2000; 432 pp; softcover
Volume: 24
ISBN-10: 0-8218-1954-2
ISBN-13: 978-0-8218-1954-8
List Price: US$128
Member Price: US$102.40
Order Code: CRMP/24
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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 étale cohomology, 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.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

Graduate students and research mathematicians interested in algebraic cycles.

Table of Contents

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