The Hodge ring of varieties in positive characteristic

Let k be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers hi,j(X)=dimHj(X,ΩXi) that hold for every smooth proper variety X over k are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of X are given by the span of the hi,0(X) and the h0,j(X) (and their duals hi,n(X) and hn,j(X)). The corresponding statements for compact Kahler manifolds were proven by Kotschick and Schreieder.