The Arithmetic and Geometry of Algebraic Cycles / Edition 1

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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 etale 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.

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Editorial Reviews

From the June 1998 Summer School come 20 contributions that explore algebraic cycles a subfield of algebraic geometry from a variety of perspectives. The papers have been organized into sections on cohomological methods, Chow groups and motives, and arithmetic methods. Some specific topics include logarithmic Hodge structures and classifying spaces; Bloch's conjecture and the -theory of projective surfaces; and torsion zero-cycles and the Abel-Jacobi map over the real numbers. Annotation c. Book News, Inc., Portland, OR
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Product Details

  • ISBN-13: 9780792361930
  • Publisher: Springer Netherlands
  • Publication date: 2/29/2000
  • Series: Nato Science Series C: (closed) , #548
  • Edition description: 2000
  • Edition number: 1
  • Pages: 615
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.44 (d)

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.

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