Algebraic Cycles and Motives
Volume 1
Edited by Jan Nagel
Edited by Chris Peters
Publisher: Cambridge University Press
Print Publication Year: 2007
Online Publication Date:May 2010
Online ISBN:9780511721496
Paperback ISBN:9780521701747
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511721496.004
Subjects: Geometry and topology, Number theory
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Introduction
This paper has two aims.
The former is to give an introduction to our earlier work and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps.
The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y. This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y.
We assume familiarity with Hodge theory and with the formalism of derived categories. On the other hand, we provide a few explicit computations of perverse truncations and intersection cohomology complexes which we could not find in the literature and which may be helpful to understand the machinery. We discuss in detail the case of surfaces, threefolds and fourfolds. In the surface case, our “intersection forms” version of the decomposition theorem stems quite naturally from two well-known and widely used theorems on surfaces, the Grauert contractibility criterion for curves on a surface and the so called “Zariski Lemma,” cf.
The following assumptions are made throughout the paper
Assumption 3.1.1.We work with varieties over the complex numbers. A map f : X → Y is a proper morphism of varieties. We assume that X is smooth. All (co)homology groups are with rational coefficients.
pp. i-iv
pp. v-viii
pp. ix-xiv
Volume 1 - Survey Articles: Read PDF
pp. 1-2
1 - The Motivic Vanishing Cycles and the Conservation Conjecture: Read PDF
pp. 3-54
2 - On the Theory of 1-Motives: Read PDF
pp. 55-101
3 - Motivic Decomposition for Resolutions of Threefolds: Read PDF
pp. 102-137
4 - Correspondences and Transfers: Read PDF
pp. 138-205
5 - Algebraic Cycles and Singularities of Normal Functions: Read PDF
pp. 206-263
6 - Zero Cycles on Singular Varieties: Read PDF
pp. 264-277
7 - Modular Curves, Modular Surfaces and Modular Fourfolds: Read PDF
pp. 278-292