Vol. 15, No. 3, 2021

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The Hodge ring of varieties in positive characteristic

Remy van Dobben de Bruyn

Vol. 15 (2021), No. 3, 729–745

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 Kähler manifolds were proven by Kotschick and Schreieder.

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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
Received: 28 January 2020
Revised: 28 June 2020
Accepted: 7 September 2020
Published: 20 May 2021
Remy van Dobben de Bruyn
Princeton University
Princeton, NJ
United States
Institute for Advanced Study
Princeton, NJ
United States