Transcendental Aspects of Algebraic Cycles

This book is based on lectures presented at the Summer School on transcendental aspects of algebraic cycles held in Grenoble. The book is divided into four chapters; the first contains introductory material on algebraic cycles, their equivalence relations, Chow varieties and consequently Lawson homology of a projective variety. The second chapter is devoted to an axiomatic set-up of Chow topology on the spaces of cycles, allowing one to distinguish homotopy type of cycle spaces and leading to the notion of Lawson homology. Also there is an explanation of morphic cohomology as a cohomology counterpart of Lawson homology and class maps into various cohomology theories. The third chapter treats from scratch classical themes of the Grothendieck theory of pure motives and standard conjectures, illustrated with many examples like the motives of curves, Picard and Albanese motives and elliptic modular motives. The definition of motivic cohomology based on the higher Chow groups is given and its various functorial properties are studied. This chapter terminates with the spectral sequence from higher Chow groups to algebraic K-theory. The last chapter contains a few lectures on the Hodge conjecture. After a general introduction, the Hodge conjecture is proved for many hyper-surfaces, e.g. the quintic fourfold. The next section of this chapter treats some infinitesimal Hodge theoretic methods with applications to the study of algebraic cycles. The last part presents applications of Hodge theoretic invariants of non-compact varieties to the study of algebraic cycles.

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