The Integral Hodge Conjecture and Universality of the Abel-Jacobi Maps

The rational Hodge conjecture states that rational Hodge classes are algebraic. This longstanding heavily studied conjecture has remained widely open since it was proposed in the nineteen fifties. In contrast, the integral Hodge conjecture is known to fail in general. To better understand the rational Hodge conjecture, it is important to ask how the integral Hodge conjecture can fail. In this thesis, we prove that there exists a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question posed by Murre on the universality of the Abel-Jacobi maps in codimension three.