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得到承认，而且在数论中也得到承认。本文的主要目的是将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上亚纯函数因子的充要条件。本文的目的是寻找一个新的与高余维代数圈有关的Hodge理论不变量，它提供了一个判定高余维代数圈有理或代数等价于零的判据.设X是一个射影光滑复簇，CH^γ（X）是X上模有理等价的余维为γ的代数圈的Chow群. Griffiths在60年代后期对上述问题取得了第一个进展，他定义了所谓的Abel-Jacobi映射ρ^γ_X：CH^γ（X）_<hom>→J^γ（X），其中CH^γ（X）_<hom>&lt;$CH^γ（X）表示同调等价于零的代数圈类的子群，J^γ（X）是复环面X的中间Jacobian。阿贝尔定理的一个解释是，如果X是黎曼曲面且γ=1，则上述映射是同构的。1968年，芒福德证明了ρ^2_X对于复曲面X一般有一个巨核，从而打破了在更一般情况下映射是同构的天真期望。本文的一个研究成果是构造了高阶Abel-Jacobi映射，它推广了Griffiths的Abel-Jacobi映射，并成功地证明了Abel-Jacobi映射核中的各种代数圈都可以被高阶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.