Degeneration of K3 surfaces with non-symplectic automorphisms

We prove that a K3 surface with an automorphism acting on the global 2 -forms by a primitive m -th root of unity, m \neq 1,2,3,4,6 , does not degenerate (assuming the existence of the so-called Kulikov models). A key result used to prove this is the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the l -adic cohomology groups of the surface.