Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i,j}\left(X\right)=dim{H}^j\left(X,{\Omega }_X^i\right)$ 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.

中文翻译：

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积极特征品种的霍奇环

让 $克$是一个具有积极特征的领域。我们证明了霍奇数之间的唯一线性关系$H^{\mathrm{一世},j}\left(X\right)=暗淡H^j\left(X,{\Omega }_X^{\mathrm{一世}}\right)$适用于每个平滑的适当品种$X$ 超过 $克$是 Serre 对偶性给出的那些。我们还表明，作为双有理不变量的霍奇数的唯一线性组合$X$ 由跨度给出 $H^{\mathrm{一世},0}\left(X\right)$ 和 $H^{0,j}\left(X\right)$ （和他们的对偶 $H^{\mathrm{一世},n}\left(X\right)$ 和 $H^{n,j}\left(X\right)$）。Kotschick 和 Schreieder 证明了紧凑 Kähler 流形的相应陈述。