Arithmetic and Geometry of Algebraic Cycles : Proceedings of the NATO Advanced Study Institute Held in Banff, Alberta, Canada, June 7-19, 1998

by Gordon, B. Brent; Lewis, James D.; Muller-Stach, Stefan; Saito, Shuji; Yui, Noriko

• ISBN13: 9780792361930
• ISBN10: 0792361938
• Format: Hardcover
• Copyright: 2000-03-01
• Publisher: Springer Verlag

Summary

The subject of algebraic cycles has thrived through its interaction with algebraic K-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology. These interactions have led to such developments as a description of Chow groups in terms of algebraic K-theory, the arithmetic Abel-Jacobi mapping, progress on the celebrated conjectures of Hodge and Tate, and the conjectures of Bloch and Beilinson. The immense recent progress in algebraic cycles, based on so many interactions with so many other areas of mathematics, has contributed to a considerable degree of inaccessibility, especially for graduate students. Even specialists in one approach to algebraic cycles may not understand other approaches well. This book offers students and specialists alike a broad perspective of algebraic cycles, presented from several viewpoints, including arithmetic, transcendental, topological, motives and K-theory methods. Topics include a discussion of the arithmetic Abel-Jacobi mapping, higher Abel-Jacobi regulator maps, polylogarithms and L-series, candidate Bloch-Beilinson filtrations, applications of Chern-Simons invariants to algebraic cycles via the study of algebraic vector bundles with algebraic connection, motivic cohomology, Chow groups of singular varieties, and recent progress on the Hodge and Tate conjectures for Abelian varieties.

Table of Contents

Preface	ix
Conference Programma	xix
Conference Picture	xxiv
List of Participants	xxv
Authors Adresses	xxix
Cohomological Methods
Lectures on algebro-geometric Chern-Weil and Cheeger-Chern-Simons theory for vector bundles	3	(26)
Bloch, Spencer
Esnault, Helene
Deligne cohomology and the geometric (co-)bar constructions	29	(22)
Gajer, Pawel
Kunga-Satake varieties and the Hodge conjecture	51	(32)
van Geemen, Bert
Hodge and Weil classes on abelian varieties	83	(34)
Murty, V. Kumar
Bloch-Kato conjecture and motivic cohomology with finite coefficients	117	(76)
Suslin, Andrei
Voevodsky, Vladimir
Chow Groups and Motives
Indecomposable hogher Chow cycles	193	(32)
Gordon, B. Brent
Lewis, James D.
Equivalence relations on algebraic Cycles	225	(36)
Jannsen, Uwe
Letter to Dick Gross on higher Abel-Jacobi maps	261	(16)
Jannsen, Uwe
Finiteness of torsion in the codimension-two Chow group: An Axiomatic Approach	277	(8)
Langer, Andreas
Algebraic cycle complexes: Basic Properties	285	(22)
Muller-Stach, Stefan
Algebraic cycles on abelian varieties: Application of abstract Fourier theory	307	(14)
Murre, Jacob P.
Motives and filtrations on Chow groups II	321	(26)
Saito, Shuji
Zero cycles on singular varieties	347	(38)
Srinivas, V.
Arithmetic Methods
Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry	385	(42)
Saito, Masa-hiko
Weight-monodromy conjecture for l-adic representations associated to modular forms: A supplement to the paper [10]	427	(6)
Saito, Takeshi
Cohomology computations related to the l-adic Abel-Jacobi map modulo l	433	(34)
Schoen, Chad
Integral elements in K-theory and products of modular curves	467	(24)
Scholl, Anthony J.
Appendix to Scholl's article: A counterexample to a conjecture of Beilinson	491	(4)
de Jeu, Rob
Reduction of abelian varieties	495	(20)
Silverberg, Alice
Zahrin, Yuri G.
The arithmetic of cetain Calabi-Yau varieties over number fields	515	(46)
Yui, Noriko
Classical and elliptic polylogarithms and special values of L-series	561
Zagier, Don
Gangl, Herbert