Rapoport-Zink空间与Kisin簇的几何

本项目主要研究Rapoport-Zink空间（简称RZ空间）以及Kisin簇的几何结构。RZ空间是p可除群的模空间，它的上同调可以实现局部Langlands对应。而Kisin簇分类的是模p的p进Galois表示对应的p进域上的有限平坦群概型的整模型。Kisin簇的连通分支集合与对应的模p的p进Galois表示的平坦形变环的连通分支集合有密切关系，Kisin利用其证明了Galois表示的模提升定理。Kisin簇的群论定义与RZ空间的群论定义极为类似，因此我们希望后者的研究方法也可以应用于前者。关于RZ空间的具体几何结构，已知结果主要集中在完全HN可分解情形，本项目研究该情形以外的一般情形，希望对于比较接近该情形的RZ空间，研究它的具体结构。关于Kisin簇的连通分支集合，现有结果集中在二维表示对应的Kisin簇，我们在本项目中希望研究高维的情形，特别是三维不可分解表示对应的Kisin簇。

In this project, we focus on the study of the geometry of Rapoport-Zink spaces and Kisin varieties. Rapoport-Zink spaces are moduli spaces of p-divisible groups. Their cohomology can be used to realize local Langlands correspondence. A Kisin variety classifies the integral models of a finite flat group scheme over a p-adic field which corresponds to a fixed mod p representation of the p-adic Galois group. The set of connected components of a Kisin variety is related to the set of connected components of the flat deformation ring of the fixed mod p Galois representation. Kisin uses the description of the set of connected components of some Kisin varieties for 2-dimensional representations to prove the modularity lifting theorem. The group theoretic definition of Kisin varieties is an inseparable analogue of the special fiber of Rapoport-Zink spaces. So we hope the methods to study the later one can also be applied to the former one. About the explicit geometry of the special fiber of the Rapoport-Zink spaces, most known results are for the fully Hodge-Newton decomposable case. We want to study other cases, especially the cases close to the fully Hodge-Newton decomposable case. For these cases, we want to give their explicit description of irreducible components. About the set of connected components of Kisin varieties, all the known results are concentrated for 2-dimensional representations. We want to study higher dimensional representations, especially 3-dimensional case.