Lectures on the Hodge and Grothendieck-Hodge conjectures (Q2898482)

The author provides an intellectually pleasing treatment of that part of Hodge theory centered around the Hodge conjecture. In a rather comprehensive, albeit user-friendly style, the reader is introduced in the first section to the Betti and de Rham cycle class maps, and the classical Hodge conjecture. The next section gives a nice treatment of Hodge loci, the deep work of Cattani, Deligne and Kaplan in this direction, absolute Hodge classes, as well as the author's approach towards reducing the classical Hodge conjecture to smooth projective varieties defined over number fields. Section three gives a thorough discussion of the (Grothendieck amended) general Hodge conjecture and example situations. Section four deals with the Bloch conjecture and some generalizations, as well as the earlier works partially leading to this conjecture due to Mumford, as well as a different approach due to Bloch (which is actually based on an earlier idea of J.-L. Colliot-Thelene). Finally, section five deals with the so-called deep nilpotent conjecture and Kimura's theorem, and how this relates to the Bloch conjecture, as well as further topics centred around the general Hodge conjecture.