Synopses & Reviews
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.
Synopsis
The NATO Advanced Study Institute on "The Arithmetic and Geometry of Algebraic Cycles" was held at the Banff Centre for Conferences in Banff (Al- berta, Canada) from June 7 until June 19, 1998. This meeting was organized jointly with Centre de Recherches Mathematiques (CRM), Montreal, as one of the CRM Summer schools which take place annually at the Banff Center. The conference also served as the kick-off activity of the CRM 1998-99 theme year on Number Theory and Arithmetic Geometry. There were 109 participants who came from 17 countries: Belgium, Canada, China, France, Germany, Greece, India, Italy, Japan, Mexico, Netherlands, - mania, Russia, Spain, Switzerland, the United Kingdom and the United States. During a period of two weeks, 41 invited lectures and 20 contributed lec- tures were presented. Four lectures by invited speakers were delivered every day, followed by two sessions of contributed talks. Many informal discussions and working sessions involving small groups were organized by individual partic- ipants. In addition, participants' reprints and preprints were displayed through- out in a lounge next to the auditorium, which further enhanced opportunities for communication and interaction.
Table of Contents
Preface. Conference Programme. Conference Picture. List of participants. Authors' addresses. Cohomological Methods. Lectures on algebro-geometric Chern-Weil and Cheeger-Chern-Simons theory for vector brundles; Bloch, et al. Deligne cohomology and the geometric (co-)bar constructions; P. Gajer. Kuga-Satake varieties and the Hodge conjecture; B. van Geemen.Hodge and Weil classes on abelian varieties; K.V. Murty. Bloch-Kato conjecture and motivic cohomology with finite coefficients; A. Suslin, V. Voevodsky. Chow Groups and Motives. Indecomposable higher Chow cycles; B.B. Gordon, J.D. Lewis. Equivalence relations on algebraic cycles; U. Jannsen. Letter to Dick Gross on higher Abel-Jacobi maps; U. Jannsen. Finiteness of torsion in the codimension-two Chow group: An Axiomatic Approach; A. Langer. Algebraic cycle complexes Basic Properties; S. Müller-Stach. Algebraic cycles on abelian varieties Application of abstract Fourier theory; J.P. Murre. Motives and filtrations on Chow groups, II; S. Saito. Zero cycles on singular varietis; V. Srinivas. Arithmetic Methods. Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry; M. Saito. Weight-monodromy conjecture for l-adic representations associated to modular forms: A supplement to the paper `IO'; T. Saito. Cohomology computations related to the l-adic Abel-Jacobi map module l; C. Schoen. Integral elements in K-theory and products of modular curves; A.J. Scholl. Appendix to Scholl's article: A counterexample to a conjecture of Beilinson; R. de Jeu. Reduction of abelian varieties; A. Silverberg, Y. Zahrin. The arithmetic of certain Calabi-You varieties over number fields; N. Yui. Classical and elliptic polylogarithms and special values of L-series; D. Zagier, H. Gangl.