Lawson同调理论与周簇的拓扑结构

We shall study the structure of projective algebraic varieties, especially singular varieties admitting the actions of the multiplicative group by using the methods in Lawson homology and Hodge theory. The problems that we shall study are the following: (1) the relations between the structures of the space of algebraic cycles on singular varieties and the structures of singular varieties themselves，including the description of singular homology、Lawson homology and their relations. (2)the algebraic and topological structure of Chow varieties. We shall continue to study important algebraic and topological invariants,including the computations of the virtual Betti numbers, Hodge numbers, Lawson homology, etc. (3) the relation between Lawson homology and Hodge structure on Abelain varieties. We shall employ new operations on Lawson homology of Abelian varieties to investigate connections between their topological filtration and Hodge structures. We have obtained interesting new results in the theory of algebraic cycles on Abelian varieties by using these operations. We expect to obtain the best upper bound of the Euler number of algebraic varieties which admit a multiplicative group action and an equivariant embedding into a complex projective space；we expect to calculate their virtual Betti numbers and Lawson homology groups, etc.； and we also expect to obtain new results related to the Friedlander-Mazur conjecture on Abelian varieties.

我们计划用Lawson同调、Hodge理论等去研究射影代数簇，特别是有乘法群作用的奇异代数簇以及阿贝尔簇等的结构。具体的问题包括：1．有乘法群作用的奇异射影代数簇上的代数cycle空间的结构与它们自身结构的联系，包括对奇异同调、Lawson同调等的刻画。2．周簇的代数和拓扑结构。进一步计算周簇的重要的代数的和拓扑的不变量，包括周簇的virtual Betti数、Hodge数、Lawson同调等。 3.阿贝尔簇的Lawson同调与Hodge结构的关系。运用阿贝尔簇的Lawson同调上的新运算去探讨拓扑滤链与Hodge结构等。利用这些运算我们对代数cycle的研究取得过有意义的成果。预期将得到有乘法群作用的代数簇在复射影空间等变嵌入时，Euler示性数的最优上界；计算周簇的virtual Betti数、Lawson同调群等；给出阿贝尔簇的Friedlander-Mazur猜想等方面的进展。

我们用Lawson同调、Hodge理论等去研究射影代数簇，特别是有乘法群作用的奇异代数簇等的结构。具体的问题包括：1．有乘法群作用的奇异射影代数簇上的代数cycle组成空间的结构与它们自身结构的联系，包括对奇异同调、Lawson同调等的刻画。2．周簇的代数和拓扑结构。我们计算出周簇的0-cycle 空间的周群同构于整数加法群。我们证明了周簇的1-cycle 空间的Lawson同调群同构于其对应的奇异同调群。我们运用复乘法群与加法群在射影代数簇上的作用，得到了这些作用下的不动点集的代数几何与拓扑不变量与射影簇的对应的不变量之间的关系。作为重要应用，我们否定了Shafarevich 关于周簇的不可约分支都是有理性的猜想。