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

(1)利用Fulton-MacPherson的双变量理论，我们构造了符号代数簇的特征类的一般理论.(2)在与J.-P.Brasselet和J.Schurmann的共同工作中，我成功地构造了奇异簇的Hirzebruch类，它是奇异Hirzebruch族的一个类.(3)利用上述(2)中得到的定理，我们得到了弦特征类和弧特征类.(4)Fulton和MacPherson猜想了二元陈氏类的存在性，1981年J.P.Brasselet在一定条件下肯定地解决了这一猜想。从那时起，它的独特性一直是一个悬而未决的问题。2002年，首席研究员横仓已经证明了它对于任何具有光滑目标的态射的唯一性。将这一结果进一步推广，与Brasselet和Schurmann一起得到了一些关于其唯一性的定理。(5)推广了上述(4)中的结果，证明了一些关于双变量特征类的定理，这可以被认为是对Fulton-MacPherson双变量理论中唯一性问题的(部分)肯定回答。(6)在构造符号代数族的特征类时，双函子起着关键作用。基于这一观察，我们提出了将Levine-Morel的代数余边定理作为双变量理论或将其推广到双变量理论，并得到了支持这一建议的一些结果。(7)首席研究员与Schurmann一起发表了一篇关于奇异空间的特征类的综述文章(约90个Oage)。(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簇特征类一般理论的特例来执行。

(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.