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

(1)。在与Lars Emstrom的合作中，我们证明了值在双变Chow群中的双变陈类的唯一存在性(2)。爆破映射是一个局部完全交态射。然而，Verdier类型的Riemann-Roch不适用于这个爆破图。在这一结果的启发下，我们还得到了其他几个结果。(3).我们发现除了Fulton和MacPherson的构造函数外，还存在着各种各样的双变量构造函数。例如，任何可构造函数本身都可以是上的双变量，而不会对其施加任何几何或拓扑条件。由此，我们指出Fulton和Macpherson在他们的书(奇异空间的研究的范畴框架)中的一个陈述是错误的，并且我们给出了它的一个修正的陈述等。我们对Victor Ginzburg在几何表示理论中引入的所谓Ginzburg-Chern类做了几点评论。受(4)中结果的启发，我们公理地证明了如果…此外，从双变可构造函数到双变同调理论都存在一个双变Chen类，如果我们将自己局限于非奇异变种的态射，则双变Chen类是唯一的，并且我们证明了它只不过是Ginzburg-Chern类。在(5)中结果的基础上，我们研究了Ginzburg-Chern类可以被捕获为双变量Chern类的态射的范畴，特别是我们讨论了非奇异簇之间的光滑态射。此外，我们还考虑了目标簇为非奇异的态射，定义了另一组比Fulton-MacPherson的双变量可构造函数更大的双变量可构造函数，并证明了从这种双变量可构造函数到双变量同调理论，Ginzburg-Chern类可以被捕获为双变量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量陈类，并且我们正在对它们进行进一步的研究。较少

(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