Studies on Characteristic Classes of Singular Varieties, Motives ans Their Related Topics

（一）.在与Lars Emstrom的合作中，我们证明了值在双变Chow群中的双变Chern类的唯一存在性（2）。一个爆破映射是一个局部完全交态射。然而，Verdier-type Riemann-Roch不适用于这个放大地图。在这个结果的激励下，我们展示了其他几个结果。（3）证明了除了富尔顿和MacPherson的构造函数外，还存在其它的双变构造函数。例如，任何可构造函数本身都可以是上的双变函数，而不需要对它施加任何几何或拓扑条件.由此指出了富尔顿和麦克弗森在《奇异空间研究的范畴框架》一书中的一个论断是错误的，并给出了一个修正的论断等.（四）、我们对维克托金兹伯格在《几何表示论》中提出的所谓金兹伯格-陈类作了几点评论。（五）、由（4）中的结果出发，我们公理化地证明了，如果 ...更多信息 从双变可构造函数到双变同调理论，存在一个双变Chen类，如果我们仅限于非奇异簇的态射，则双变Chen类是唯一的，并进一步证明了它只是Ginzburg-Chern类。（六）、基于（5）中的结果，我们研究了Ginzburg-Chern类可以被捕获为双变Chern类的态射范畴，特别是我们处理了非奇异簇之间的光滑态射.（7）.进一步考虑目标簇为非奇异的态射，定义了另一组比Fulton-MacPherson双变可构造函数更大的双变可构造函数，并证明了Ginzburg-Chern类可以从双变可构造函数到双变同调理论被捕获为双变Chern类。在任意态射的情况下，受（7）中所得结果的启发，我们引入了形式双变陈类，并进一步研究它们。少

(1). In a joint work with Lars Emstrom we showed the unique existence of the bivariant Chern class with values in the bivariant Chow groups(2). A blow-up map is a local complete intersection morphism. However, a Verdier-type Riemann-Roch does not hold for this blow-up map. And motivated by this result we showed several other otherresults.(3).We obserbved that there exist various bivariant constructive functions other than those of Fulton and MacPherson. For example, any constructible function itself can be a bivariant on without imposing any geometric or topological condition on it. With this we point out that a statement made by Fulton and macPherson in their book (Categorical frameworks for the study of singular spaces) is false and furthermore we gave a modified statement of it and etc.(4). We made several remarks on the so-called Ginzburg-Chern class introduced by Victor Ginzburg in Geometric Representation Theory.(5). Moivated by the results in (4), we showed axiomatically that if … More there exist a bivariant chern class from the bivariant constructible function to the bivariant homology theory and if we restrict ourselves to the morphisms to nonsingular varieties the bivariant Chen class is unique and furthermore we showed that it is nothing but the Ginzburg-Chern class.(6). Based on the results in (5), we investigated categories of morphisms in which the Ginzburg-Chern class can be captured as a bivariant Chern class, in particular we treated smooth morphisms between nonsingular varieties.(7).Furthermore we consider morphisms with target varieties being nonsingular, and we defined another group of bivariant consructible functions which is larger than that of Fulton-MacPherson's bivariant constructible functions and we showed that the Ginzburg-Chern class can be captured as a bivariant Chern class from this bivariant constructible function to the bivariant homology theory. And in the case of arbitrary morphisms, motivated by the results obtsained in (7), we introduced formal bivariant Chern classes and we are investigating them further. Less