Algebraic Cycles on Algebraic Varieties

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The history of the study of algebraic cycles is long and its significance is recognized not only in algebraic geometry but also in number theory. The main purpose of the reaserch is to generalize Abel's theorem to the higher dimensional case. Abel's theorem gives the neccesary and sufficient condition for a divisor on a Riemann surface X to be the divisor of a meromorphic function on X. The aim of the reserach is to find a new Hodge theoretic invariant associated to an algebraic cycle of higher codimension which provides a criterion of the cycle to be rationally, or algebraically equivalent to zero.Let X be a projective smooth complex variety and let CH^γ(X) be the Chow gropup that is the group of algebraic cycles of codimension γ on X modulo rational equivalence. The first progress toward the above problem was made by Griffiths in the late 60th when he defined the so-called Abel-Jacobi map ρ^γ_X:CH^γ(X)_<hom>→J^γ(X) where CH^γ(X)_<hom> ⊂ CH^γ(X) denotes the subgroup of the classes of those algebraic cycles which are homologically equivalent to zero and J^γ(X) is the intermediate Jacobian of X which is a complex torus. A paraphrase of the Abel's theorem is that the above map is an isomorphism if X is a Riemann surface and γ=1. The naive expectation that the map would be an isomorphism in more general cases was blown out in 1968 when Mumford proved that ρ^2_X has in general a gigantic kernel for a complex surface X. A fruit of this research project is the construction of higher Abel-Jacobi map, which generalizes Griffiths Abel-Jacobi map and succeeded in showing that various algebaic cycles in the kernel of Abel-Jacobi map can be captured by higher Abel-Jacobi map. It indicates that the higher Abel-jacobi map is bringing out a new perspective in the study of algebraic cycles.