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The arithmetic and geometry of algebraic cycles. Proceedings of the NATO Advanced Study Institute, Banff, Canada, June 7–19, 1998

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Notes for a mini course at the University of Science and Technology of China in Hefei, China, June 23–July 12, 2014.
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These notes form an expanded version of some introductory lec-tures to be delivered at the Workshop on Arithmetic and Geometry of K3 surfaces and Calabi-Yau Threefolds, August 16-25, 2011, at the Fields Insti-tute. After presenting a general overview, we begin with some rudimentary aspects of Hodge theory and algebraic cycles. We then introduce Deligne co-homology, as well as generalized cycles that are connected to higher K-theory, and associated regulators. Finally, we specialize to the Calabi-Yau situation, and explain some recent developments in the field.
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We construct some natural indecomposable elements of \(CH^g(J(C),1)\) with trivial regulator, and in particular, prove that \(\) is uncountable for C a generic curve or a generic hyperelliptic curve of genus \(g \geq 3\).
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