仿射概型上Euler类群理论研究

The Euler Class Group Theory, which has its genesis in topology, was outlined by M. V. Nori around 1990 in Chicago University, aimed to develop an obstruction theory for algebraic vector bundles over affine varieties. Now, after a near 20 years' development, this theory not only give a fully settlement to the obstruction problem of algebraic vector bundles with top rank over affine varieties, but also became one of the most important theories in the related subjects of projective modules and algebraic vector bundles in recent history. In this task, our researching aim is to further develop Euler Class Group Theory. We hope that our works could connect this theory with K-theory and Chow ring of the variety, which are the most classical two homology theories about algebraic vector bundles. We plan to extend the Riemann-Roch theorem to the case of Generalized Euler Class Group. And beside this, we would also try to prove some basic results about Chow ring, such as Homology Sequences and Projective Formula, in the case of Euler Class Groups. In this task, we also plan to do some works about the extension of the Euler Class Group Theory of affine schemes to more general schemes. By this research, we want to extend the related results about algebraic vector bundles over affine scheme, such as algebraic obstruction theory and Riemann-Roch theorem, to the case of the algebraic vector bundles over more general schemes.

为了建立仿射代数簇上代数矢量丛的障碍理论，上世纪90年代芝加哥大学教授M. V. Nori提出了Euler 类群的理论构想。进过近二十年的发展，如今这一理论不但成功的解决了仿射代数簇上满秩矢量丛的障碍问题，而且业已成为代数矢量丛、投射模等相关领域中极为重要的理论之一。 本项目的研究旨在深入探索该理论同其他经典理论间的联系。我们的工作目标之一是构建Euler类群理论与另外两个有关矢量丛的经典同调理论："K-理论和Chow环理论，"之间的联系。我们将研究推广有关Euler类群的Riemann-Roch定理到一般Euler类群上。同时，我们将探索在Euler类群的理论框架下证明有关Chow环的正合序列、投影公式等基本结果。本项目中，我们还将开展针对更一般概型上Euler类群理论的研究，希望建立针对一般概型上代数矢量丛的障碍理论以及Riemann-Roch定理等结果。