We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\bar\rho$ of a $p$-adic field $K$ and a cocharacter $\mu$. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied with $n=3$ or $\mu$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\bar\rho$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general. 

　　Publication: 

　　Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 785 

　　Author: 

　　Miaofen Chen 

　　Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, No. 500, Dong Chuan Road, Shanghai 200241, China 

　　E-mail: mfchen@math.ecnu.edu.cn 

　　Sian Nie 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Beijing 100190, China 

　　E-mail: niesian@amss.ac.cn