The Hodge ring of varieties in positive characteristic

Abstract

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i,j}\left(X\right)=dim{H}^j\left(X,{\mathrm{\Omega }}_X^i\right)$ 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 $h^{i,0}\left(X\right)$ and the $h^{0,j}\left(X\right)$ (and their duals $h^{i,n}\left(X\right)$ and $h^{n,j}\left(X\right)$). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.