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

点击翻译按钮获取中文摘要

(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