Research on non-abelian Hodge structures of algebraic varieties

1)研究正特征情况下的非阿贝尔Hodge分解(或过滤)；2)研究非对易方案的形式化理论与非阿贝尔Hodge理论之间的关系；3)研究代数族的非阿贝尔混合Hodge结构。对于第一个目的，我意识到利用n-堆栈构造晶体同伦理论是很重要的。如果建立了这样的理论，原来的问题就会变得清晰，就会有很多应用。这将是N.Katz在阿贝尔情形下关于Frobenius映射和Hodge滤子的工作的推广。我现在正在尝试下一个大项目。关于第二个问题，在研究了非阿贝尔格式之后，我了解到，将变形参数空间描述为n-堆栈更为自然。这种描述将有助于研究与非阿贝尔Hodge结构的关系。在第三个方面，辛普森有了很大的发展。他将非交换混合Hodge结构描述为非交换上同调n-堆栈上的滤子。我研究了他的理论，并对n-栈上的这种过滤进行了一些显式计算。此外，我还与Y.Takeda一起研究了探戈之前的曲线结构。这项工作与第一个有关。与K.Ohba一起，我研究了Gerbs(使用n-叠层)的研究和模空间循环的研究之间的关系。

There were the following three purposes for this project.1) Study of the non-abelian Hodge decomposition (or filtration) in the case of positive characteristic.2) Study of the relation between defomation theory of non-commutative schemes and non-abelian Hodge theory.3) Study of non-abelian mixed Hodge structures of algebraic varieties.As for the first purpose, I realized that it would be important to construct the crystellin homotopy theory using n-stacks. If such a theory is constructed, the original problem would be clear and there would be many applications. It would be an extention of the work of N. Katz on Frobenius maps and Hodge filtrations in the abelian cases. I am now trying this next big project. On the second one, after the study of non-abelian schemes I understood that it is more natural to describe the deformation parameter space as an n-stacks. Such description would be useful to study the relationship with non-abelian Hodge structures. On the third one, there was a great development by C. Simpson. He describe the non-abelian mixed Hodge structures as kinds of filtrations on non-abelian cohomology n-stacks. I studied his theory and make some explicit calculations of such filtrations on n-stacks. Moreover I worked with Y. Takeda on the pre-Tango structures on curves. The work is related on the first one. With K. Ohba, I studied on the relation between the study of gerbs (using n-stacks) and the study of cycles of moduli spaces.