A general study on topology, analysis and geometry of symbolic algebraic varieties

(1)利用Fulton-MacPherson的双变理论，构造了符号代数变体特征类的一般理论。(2)与j.p。Brasselet和J. Schurmann I成功地构造了Hirzebruch奇异变种类，这是Hirzebruch奇异属的一个类版本。(3)利用上(2)所得到的定理，我们得到了弦特征分类和弧特征分类，它们是弦陈氏类和弧欧拉特征的推广。(4) Fulton和MacPherson在1981年推测了双变Chern类的存在。在一定条件下，布拉塞特肯定地解决了这个问题。从那时起，它的独特性就成了一个悬而未决的问题。2002年，首席研究员Yokura已经证明了它在光滑目标上的任何形态变异的独特性。在此基础上，主要研究者与Brasselet和Schurmann一起，得到了关于其唯一性的定理。(5)推广上(4)的结果，证明了一些关于双变特征类的定理，这些定理可以看作是Fulton-MacPherson双变理论中唯一性问题的一个（偏）正答案。(6)在符号代数变体的特征类的构造中，一个基本的问题是双分子起着关键的作用。在此基础上，我们提出将Levine-Morel的代数协共论作为一种双变理论或将其扩展为一种双变理论，并得到了一些结果来支持这一建议。(7)与Schurmann共同发表了一篇关于奇异空间特征类的综述文章（约90页）。(8)不幸的是，我们不能将捕获等变特征类的程序作为符号代数变体特征类的一般理论的特例。

(1) We constructed a general theory of characteristic classes of symbolic algebraic varieties, using Fulton-MacPherson's bivariant theory.(2) In a joint work with J.-P. Brasselet and J. Schurmann I succeeded in the construction of Hirzebruch classes of singular varieties, which is a class version of the singular Hirzebruch genera.(3) Using the theorems obtained in the above (2), we obtained stringy characteristic classed and arc characteristic classes, which are a generalization of the stringy Chern class and arc Euler chracteristic.(4) The existence of a bivariant Chern class was conjectured by Fulton and MacPherson and in 1981J.-P. Brasselet solved it affirmatively under a certain condition. Since then, its uniqueness has been an open problem. In 2002 the head investigator, Yokura, has shown its uniqueness for any morphism with the smooth target. Extending this result furthermcre, the head investigator, together with Brasselet and Schurmann, obtained some theorems concerning its uniqueness.(5) Generalizing the results obtained in the above (4), we proved some theorems concerning bivariant characteristic classes, which may be considered as a (poartial) positive answer to a uniqueness problem in Fulton-MacPherson's bivariant theory.(6) A fundamental thing in the construction of characterisic classes of symbolic algebraic varieties is that bifunctors play a key role. Based on this observation, we proposed to capture Levine-Morel's algebraic cobordism as a bivariant theory or to extend it to a bivariant theory, and we obtained some results to support this proposal.(7) The head investigator, together with Schurmann, published a survey article of characteristic classes of singular spaces (about 90 oages).(8) Unfortuntely we could not carry out a program of capturing equivariant characteristic classes as a special case of a general theory of characteristic classes of symbolic algebraic varieties.