Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

Abstract

We consider the Kisin variety associated to a n-dimensional absolutely irreducible mod p Galois representation ρ of a p-adic field K and a cocharacter µ. 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 µ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to ρ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.