Chow groups of large coniveau complete intersections

This chapter first describes how to compute the Hodge coniveau of complete intersections. It then explains a strategy to attack the generalized Hodge conjecture for complete intersections of coniveau 2. The guiding idea is that although the powerful method of the decomposition of the diagonal suggests that computing Chow groups of small dimension is the right way to solve the generalized Hodge conjecture, it might be better to invert the logic and try to compute the geometric coniveau directly. And indeed, this chapter culminates with the proof of the fact that for very general complete intersections, the generalized Hodge conjecture implies the generalized Bloch conjecture.

Review of Hodge theory and algebraic cycles

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.

Refined Motivic Dimension

We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.

Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes

Suppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D and that U is the complement of the ramification locus in Y. The first theorem in this paper implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance, this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to more general nonabelian covers.