Cohomology theory in birational geometry

This is a continuation of [9], where it was shown that K-equivalent complex projective manifolds have the same Betti numbers by, using the theory of p-adic integrals and Deligne's solution to the Weil conjecture. The aim of this note is to show that with a little more book-keeping work, namely by applying Faltings' p-adic Hodge Theory, our p-adic method also leads to the equivalence of Hodge numbers - a result which was previously known via motivic integration.