Softcover ISBN:  9780821819548 
Product Code:  CRMP/24 
432 pp 
List Price:  $136.00 
MAA Member Price:  $122.40 
AMS Member Price:  $108.80 
Electronic ISBN:  9781470439385 
Product Code:  CRMP/24.E 
432 pp 
List Price:  $128.00 
MAA Member Price:  $115.20 
AMS Member Price:  $102.40 

Book DetailsCRM Proceedings & Lecture NotesVolume: 24; 2000MSC: Primary 11; 14;
The NATO ASI/CRM Summer School at Banff offered a unique, full, and indepth 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 MerkurjevSuslin theorem to the arithmetic AbelJacobi 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 nonzero 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.ReadershipGraduate students and research mathematicians interested in algebraic cycles.

Table of Contents

Cohomological Methods

Filtrations on the cohomology of abelian varieties

Building mixed Hodge structures

The AtiyahChern character yields the semiregularity map as well as the infinitesimal AbelJacobi map

Regulators and characteristic classes of flat bundles

Height pairings asymptotics and BottChern forms

Logarithmic Hodge structures and classifying spaces

Chow Groups and Motives

Motives and algebraic de Rham cohomology

Hermitian vector bundles and characteristic classes

The mixed motive of a projective variety

Bloch’s conjecture and the $K$theory of projective surfaces

From Jacobians to onemotives: Exposition of a conjecture of Deligne

Motives, algebraic cycles and Hodge theory

Arithmetic methods

PicardFuchs uniformization: Modularity of the mirror map and mirrormoonshine

Hilbert modular varieties in positive characteristic

On the NéronSeveri groups of some $K$3 surfaces

Torsion zerocycles and the AbelJacobi map over the real numbers

A remark on the Griffiths groups of certain product varieties

$p$adic AbelJacobi maps and $p$adic heights

Crystalline fundamental groups and $p$adic Hodge theory

Thompson series, and the mirror maps of pencils of $K$3 surfaces


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 Table of Contents

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The NATO ASI/CRM Summer School at Banff offered a unique, full, and indepth 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 MerkurjevSuslin theorem to the arithmetic AbelJacobi 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 nonzero 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.
Graduate students and research mathematicians interested in algebraic cycles.

Cohomological Methods

Filtrations on the cohomology of abelian varieties

Building mixed Hodge structures

The AtiyahChern character yields the semiregularity map as well as the infinitesimal AbelJacobi map

Regulators and characteristic classes of flat bundles

Height pairings asymptotics and BottChern forms

Logarithmic Hodge structures and classifying spaces

Chow Groups and Motives

Motives and algebraic de Rham cohomology

Hermitian vector bundles and characteristic classes

The mixed motive of a projective variety

Bloch’s conjecture and the $K$theory of projective surfaces

From Jacobians to onemotives: Exposition of a conjecture of Deligne

Motives, algebraic cycles and Hodge theory

Arithmetic methods

PicardFuchs uniformization: Modularity of the mirror map and mirrormoonshine

Hilbert modular varieties in positive characteristic

On the NéronSeveri groups of some $K$3 surfaces

Torsion zerocycles and the AbelJacobi map over the real numbers

A remark on the Griffiths groups of certain product varieties

$p$adic AbelJacobi maps and $p$adic heights

Crystalline fundamental groups and $p$adic Hodge theory

Thompson series, and the mirror maps of pencils of $K$3 surfaces