On the failure of the integral Tate conjecture for products with projective hypersurfaces

Kok, Kees

doi:10.2140/ant.2026.20.119

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On the failure of the integral Tate conjecture for products with projective hypersurfaces

Kees Kok

Vol. 20 (2026), No. 1, 119–146

DOI: 10.2140/ant.2026.20.119

Abstract

We show the failure of the integral Tate conjecture for the product of a smooth odd-dimensional projective hypersurface with certain smooth projective varieties. To do this, we use a similar specialization argument developed by Gabber (2002) and Colliot-Thélène (2019), now applied to Schreieder’s refined unramified cohomology (2023). The results thus obtained give an interpretation of Shen’s result (2021) in terms of refined unramified cohomology. We avoid the need to work over the complex numbers and in turn, all results hold over general algebraically closed fields of characteristic not 2.

Keywords

integral Tate conjecture, refined unramified cohomology, étale cohomology, blow-up, Lefschetz pencil, specialization, cospecialization, Poincaré duality

Mathematical Subject Classification

Primary: 14C21, 14C25, 14D06, 14F20

Milestones

Received: 15 December 2023

Revised: 25 November 2024

Accepted: 20 January 2025

Published: 28 November 2025

Authors

Kees Kok
KdV Institute for Mathematics University of Amsterdam 1098 XG Amsterdam Netherlands

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Vol. 20, No. 1, 2026