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We consider the Kisin variety associated to an  n -dimensional absolutely irreducible mod  p  Galois representationρ  ¯{\bar{\rho}}    of a  p -adic field  K  together with 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 ramified withn  =  3{n=3}    or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated toρ  ¯{\bar{\rho}}    of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.

Connectedness of Kisin varieties associated to absolutely irreducible Galois representations