**CRM Proceedings & Lecture Notes**

Volume: 24;
2000;
432 pp;
Softcover

MSC: Primary 11; 14; 19;

Print ISBN: 978-0-8218-1954-8

Product Code: CRMP/24

List Price: $128.00

Individual Member Price: $102.40

**Electronic ISBN: 978-1-4704-3938-5
Product Code: CRMP/24.E**

List Price: $128.00

Individual Member Price: $102.40

# The Arithmetic and Geometry of Algebraic Cycles

Share this page *Edited by *
*B. Brent Gordon; James D. Lewis; Stefan Müller-Stach; Shuji Saito; Noriko Yui*

A co-publication of the AMS and Centre de Recherches Mathématiques

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

## The Arithmetic and Geometry of Algebraic Cycles

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Contents of the Proceedings of the NATO Advance Study Institute: Arithmetic and geometry of algebraic cycles xv16
- Conference programme xvii18
- Photo xxi22
- List of participants xxiii24
- Authors’ addresses xxvii28
- Cohomological Methods 130
- Filtrations on the cohomology of abelian varieties 332
- Building mixed Hodge structures 1342
- The Atiyah-Chern character yields the semiregularity map as well as the infinitesimal Abel-Jacobi map 3362
- Regulators and characteristic classes of flat bundles 4776
- Height pairings asymptotics and Bott-Chern forms 93122
- Logarithmic Hodge structures and classifying spaces 115144
- Chow Groups and Motives 131160
- Motives and algebraic de Rham cohomology 133162
- Hermitian vector bundles and characteristic classes 155184
- The mixed motive of a projective variety 183212
- Bloch’s conjecture and the 𝐾-theory of projective surfaces 195224
- From Jacobians to one-motives: Exposition of a conjecture of Deligne 215244
- Motives, algebraic cycles and Hodge theory 235264
- Arithmetic methods 255284
- Picard-Fuchs uniformization: Modularity of the mirror map and mirror-moonshine 257286
- Hilbert modular varieties in positive characteristic 283312
- On the Néron-Severi groups of some 𝐾3 surfaces 305334
- Torsion zero-cycles and the Abel-Jacobi map over the real numbers 329358
- A remark on the Griffiths groups of certain product varieties 361390
- 𝑝-adic Abel-Jacobi maps and 𝑝-adic heights 367396
- Crystalline fundamental groups and 𝑝-adic Hodge theory 381410
- Thompson series, and the mirror maps of pencils of 𝐾3 surfaces 399428
- Back Cover Back Cover1462