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Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)$
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Claire Voisin

Print publication date: 2014

Print ISBN-13: 9780691160504

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691160504.001.0001

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Review of Hodge theory and algebraic cycles

Review of Hodge theory and algebraic cycles

(p.15) Chapter Two Review of Hodge theory and algebraic cycles
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

Claire Voisin

Princeton University Press

This chapter provides the background for the studies to be undertaken in succeeding chapters. It reviews Chow groups, correspondences and motives on the purely algebraic side, cycle classes, and (mixed) Hodge structures on the algebraic–topological side. Emphasis is placed on the notion of coniveau and the generalized Hodge conjecture which states the equality of geometric and Hodge coniveau. The chapter first follows the construction of Chow groups, the application of the localization exact sequence, the functoriality and motives of Chow groups, and cycle classes. It then turns to Hodge structures; pursuing related topics such as polarization, Hodge classes, standard conjectures, mixed Hodge structures, and Hodge coniveau.

Keywords:   Chow groups, Hodge structures, cycle classes, coniveau, generalized Hodge conjecture, Hodge classes, mixed Hodge structures, functoriality

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