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 $h^{i,j}(X)=\dim H^j(X,{\Omega }_X^i)$ 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 $h^{i,0}(X)$ and the $h^{0,j}(X)$ (and their duals $h^{i,n}(X)$ and $h^{n,j}(X)$). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.

16 pages

References in corpus (2)

• The construction problem for Hodge numbers modulo an integer
• The construction problem for Hodge numbers modulo an integer in positive characteristic

Cited by in corpus (2)

• The construction problem for Hodge numbers modulo an integer in positive characteristic
• Hodge numbers are not derived invariants in positive characteristic