Title Algebraic cycles and motives. Volume 1 / edited by Jan Nagel, Chris Peters

Published Cambridge ; New York : Cambridge University Press, 2007

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Description 1 online resource (xiv, 292 pages) : illustrations Series London Mathematical Society lecture note series ; 343 London Mathematical Society lecture note series ; 343. Contents Cover; Title; Copyright; Contents of Volume 1; Preface; Volume 1: Survey Articles ; 1 The Motivic Vanishing Cycles and the Conservation Conjecture ; 1.1 Introduction; 1.2 The classical pictures; 1.2.1 The vanishing cycles formalism in étale cohomology; 1.2.2 The Rapoport-Zink construction; 1.2.3 The limit of a variation of Hodge structures; 1.2.4 The analogy between the situations in étale cohomology and Hodge theory; 1.3 Specialization systems; 1.3.1 The motivic categories 1.3.2 Definitions and examples1.3.3 The basic results; 1.4 Constructing the vanishing cycles formalism; 1.4.1 The idea of the construction; 1.4.2 The cosimplicial motive A and the construction of Y; 1.4.3 The construction of \; 1.4.4 Pseudo-monoidal structure; 1.4.5 Compatibility with duality; 1.4.6 The monodromy operator; 1.5 Conservation conjecture. Application to Schur finiteness of motives; 1.5.1 The statement of the conjecture; 1.5.2 About the Schur finiteness of motives; 1.5.3 The conservation conjecture implies the Schur finiteness of motives 1.5.4 Some steps toward the Conservation conjecture2 On the Theory of 1-Motives ; 2.1 On Picard Functors; 2.1.1 Simplicial Picard Functors; 2.1.2 Relative Picard Functors; 2.1.3 Higher Picard Functors; 2.2 On 1-Motives; 2.2.1 Generalities; 2.2.2 Hodge Realization; 2.2.3 Flat, ℓ-adic and Étale Realizations; 2.2.4 Crystalline Realization; 2.2.5 De Rham Realization; 2.2.6 Paradigma; 2.2.7 Cartier Duality; 2.2.8 Symmetric Avatar; 2.2.9 1-Motives with Torsion; 2.2.10 1-Motives up to Isogenies; 2.2.11 Universal Realization and Triangulated 1-Motives 2.2.12 1-Motives with Additive Factors2.3 On 1-Motivic (Co)homology; 2.3.1 Albanese and Picard 1-Motives; 2.3.2 Hodge 1-Motives; 2.3.3 Non-homotopical Invariant Theories; 2.3.4 Final Remarks; 3 Motivic Decomposition for Resolutions of Threefolds ; 3.1 Introduction; 3.2 Intersection forms; 3.2.1 Surfaces; 3.2.2 Intersection forms associated to a map; 3.2.3 Resolutions of isolated singularities in dimension 3; 3.3 Intersection forms and Decomposition in the Derived Category; 3.3.1 Resolution of surface singularities 3.3.2 Fibrations over curves3.3.3 Smooth maps; 3.4 Perverse sheaves and the Decomposition Theorem; 3.4.1 Truncation and Perverse sheaves; 3.4.2 The simple objects of P(Y); 3.4.4 The Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber; 3.4.5 Results on intersection forms; 3.4.6 The decomposition mechanism; 3.5 Grothendieck motive decomposition for maps of threefolds; 4 Correspondences and Transfers ; 4.1 Finite Correspondences; 4.1.1 Relative Cycles; 4.1.2 Composition of Finite Correspondences; 4.1.3 Monoidal Structure Summary 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 Bibliography Includes bibliographical references Notes English Print version record Subject Algebraic cycles -- Congresses Motives (Mathematics) -- Congresses MATHEMATICS -- Geometry -- Algebraic. Algebraic cycles Motives (Mathematics) Genre/Form Conference papers and proceedings Form Electronic book Author Nagel, Jan. Peters, C. (Chris) Murre, Jacob P., 1929- European Algebraic Geometry Research Training Network ISBN 9781107362987 1107362989 9780511893926 0511893922 9780511721496 0511721498 0521701740 9780521701747 1139882708 9781139882705 1107367891 9781107367890 1107372437 9781107372436 1107369924 9781107369924 1299405495 9781299405493 1107365430 9781107365438

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