Algebraic Cycles on Algebraic Varieties

代数圈的研究历史悠久，它的重要性不仅在代数几何中被认识到，而且在数论中也被认识到。研究的主要目的是将Abel定理推广到高维情形。Abel定理给出了黎曼曲面X上的因子是X上的亚纯函数的因子的充要条件.研究的目的是找到一个与高余维代数圈相关的新的Hodge理论不变量，它给出了这个圈有理或代数等价于零的判据.设X是射影光滑复簇，CH^γ(X)是X上模有理等价的余维γ的代数圈群.上述问题的第一个进展是Griffiths在60年代末定义了所谓的Abel-Jacobi映射ρ^γ_X：CH^γ(X)_&lt；→J^γ(X)，其中CH^γ(X)_&lt；hom&gt；⊂CH^γ(X)表示那些同调等价于零的代数圈类的子群，J^γ(X)是X的中间雅可比。Abel定理的一种解释是，如果X是Riemann曲面，且γ=1，则上述映射是同构的。1968年，当Mumford证明ρ^2_X对于复杂曲面X具有一般的巨核时，关于映射在更一般情况下是同构的天真期望被打破。这项研究的一个成果是构造了高阶Abel-Jacobi映射，它推广了Griffiths Abel-Jacobi映射，并成功地证明了Abel-Jacobi映射核中的各种代数循环可以被高阶Abel-Jacobi映射捕获。这表明高阶Abel-Jacobi映射为代数圈的研究带来了新的视角。

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.