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

(1)利用Fulton-MacPherson的双变理论，我们构造了符号代数簇的特征类的一般理论。(2)在与J的合作中，P. Schulelet和J. Schurmann I成功地构造了奇异变种的Hirzebruch类，这是奇异Hirzebruch属的一个类版本。(3)利用（2）中得到的定理，我们得到弦特征类和弧特征类，它们是弦Chern类和弧Euler特征类的推广。(4)双变陈类的存在性由富尔顿和麦克弗森证明，并于1981年J. P. Eschelet在一定条件下肯定地解决了这个问题。从那时起，它的独特性一直是一个开放的问题。在2002年，首席研究员，Yokura，已经证明了它的唯一性的任何态射与光滑的目标。进一步推广这一结果，首席研究员，连同Atrielet和Schurmann，获得了一些定理，关于它的唯一性。(5)推广了（4）中的结果，证明了关于双变特征类的几个定理，这些定理可被认为是对Fulton-MacPherson双变理论中一个唯一性问题的（极）肯定回答。(6)在符号代数簇的特征类的构造中，双函子起着关键的作用。在此基础上，我们提出了将Levine-Morel的代数配边理论作为一个双变理论或将其推广到一个双变理论，并得到了一些结果来支持这一建议. (7)首席调查员，连同舒曼，发表了一篇调查文章的特征类奇异空间（约90 oages）。(8)不幸的是，我们不能进行一个程序，捕捉等变的特征类作为一个特殊情况下的一般理论的特征类的符号代数簇。

(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.