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.
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.