The Hodge ring of varieties in positive characteristic

van Dobben de Bruyn, Remy

doi:10.2140/ant.2021.15.729

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The Hodge ring of varieties in positive characteristic

Remy van Dobben de Bruyn

Vol. 15 (2021), No. 3, 729–745

DOI: 10.2140/ant.2021.15.729

Abstract

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

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Keywords

algebraic geometry, positive characteristic, Hodge cohomology, de Rham cohomology, Grothendieck ring of varieties, birational invariants

Mathematical Subject Classification 2010

Primary: 14G17

Secondary: 14A10, 14E99, 14F40, 14F99

Milestones

Received: 28 January 2020

Revised: 28 June 2020

Accepted: 7 September 2020

Published: 20 May 2021

Authors

Remy van Dobben de Bruyn
Princeton University Princeton, NJ United States	Institute for Advanced Study Princeton, NJ United States

Vol. 15, No. 3, 2021