Algebraic Cycles and Motives: Volume 1 (London Mathematical Society Lecture Note Series, Series Number 343)

Algebraic geometry is a central subfield of mathematics in which the study of cycles is an important theme. Alexander Grothendieck taught that algebraic cycles should be considered from a motivic point of view and in recent years this topic has spurred a lot of activity. This 2007 book is one of two volumes that provide a self-contained account of the subject. Together, the two books contain twenty-two contributions from leading figures in the field which survey the key research strands and present interesting new results. Topics discussed include: the study of algebraic cycles using Abel-Jacobi/regulator maps and normal functions; motives (Voevodsky's triangulated category of mixed motives, finite-dimensional motives); the conjectures of Bloch-Beilinson and Murre on filtrations on Chow groups and Bloch's conjecture. Researchers and students in complex algebraic geometry and arithmetic geometry will find much of interest here. Part I. Survey Articles: The Motivic Vanishing Cycles And The Conservation Conjecture / J. Ayoub ; On The Theory Of 1-motives / L. Barbieri-viale ; Motivic Decomposition For Resolutions Of Threefolds / M. De Cataldo And L. Migliorini ; Correspondences And Transfers / F. Déglise ; Algebraic Cycles And Singularities Of Normal Functions / M. Green And / Ph. Griffiths ; Zero Cycles On Singular Varieties / A. Krishna And / V. Srinivas ; Modular Curves, Modular Surfaces And Modular Fourfolds / D. Ramakrishnan. Part Ii. Research Articles: Beilinson's Hodge Conjecture With Coefficients / M. Asakura And / S. Saito ; On The Splitting Of The Bloch-beilinson Filtration / A. Beauville ; Künneth Projectors / S. Bloch And H. Esnault ; The Brill-noether Curve Of A Stable Bundle On A Genus Two Curve / S. Brivio And A. Verra ; On Tannaka Duality For Vector Bundles On P-adic Curves / C. Deninger And A. Werner ; On Finite-dimensional Motives And Murre's Conjecture / U. Jannsen ; On The Transcendental Part Of The Motive Of A Surface / B. Kahn, J.p. Murre And C. Pedrini ; A Note On Finite Dimensional Motives / S.i. Kimura ; Real Regulators On Milnor Complexes, Ii / J.d. Lewis ; Motives For Picard Modular Surfaces / A. Miller [and Others] ; The Regulator Map For Complete Intersections / J. Nagel ; Hodge Number Polynomials For Nearby And Vanishing Cohomology / C. Peters And J. Steenbrink ; Direct Image Of Logarithmic Complexes / M. Saito ; Mordell-weil Lattices Of Certain Elliptic K3's / T. Shioda ; Motives From Diffraction / J. Stienstra. Edited By Jan Nagel, Chris Peters. These Proceedings Contain A Selection Of Papers From The Eager Conference 'algebraic Cycles And Motives' That Was Held At The Lorentz Center In Leiden On The Occasion Of The 75th Birthday Of Professor J.p. Murre (aug 30-sept 3, 2004)--pref. Includes Bibliographical References. Contents Preface Volume 1: Survey articles 1. The Motivic Vanishing Cycles and the Conservation Conjecture 2. On the Theory of 1-Motives 3. Intersection Forms, Topology of Maps and Motivic Decomposition for Resolutions of Threefolds 4. Finite Correspondences and Transfers over a Regular Base 5. Algebraic Cycles and Singularities of Normal Functions 6. Zero Cycles on Singular Varieties 7. Modular Curves, Modular Surfaces, and Modular Fourfolds These two volumes provide a self-contained account of research on algebraic cycles and motives. Twenty-two contributions from leading figures survey the key research strands, including: Abel-Jacobi/regulator maps and normal functions; Voevodsky's triangulated category of mixed motives; conjectures of Bloch-Beilinson and Murre on filtrations on Chow groups.