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TORSION BIRATIONAL MOTIVES OF SURFACES AND UNRAMIFIED COHOMOLOGY

Part of: Cycles and subschemes $K$-theory in geometry Special varieties

Published online by Cambridge University Press:  21 July 2025

Kanetomo Sato Open the ORCID record for Kanetomo Sato [Opens in a new window]  and
Takao Yamazaki
Kanetomo Sato*
Affiliation:
Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
Takao Yamazaki
Affiliation:
Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan (ytakao@math.chuo-u.ac.jp)
*
kanetomo@math.chuo-u.ac.jp
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Abstract

Let S and T be smooth projective varieties over an algebraically closed field k. Suppose that S is a surface admitting a decomposition of the diagonal. We show that, away from the characteristic of k, if an algebraic correspondence $T \to S$ acts trivially on the unramified cohomology, then it acts trivially on any normalized, birational and motivic functor. This generalizes Kahn’s result on the torsion order of S. We also exhibit an example of S over $\mathbb {C}$ for which $S \times S$ violates the integral Hodge conjecture.

Keywords

Unramified cohomologybirational motivesdecomposition of diagonal

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14M20: Rational and unirational varieties 19E15: Algebraic cycles and motivic cohomology

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 24 , Issue 6 , November 2025 , pp. 2283 - 2315
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

In memory of Noriyuki Suwa

The first author is supported by JSPS KAKENHI Grant (JP20K03566). The second author is supported by JSPS KAKENHI Grant (JP21K03153) and by Chuo University Grant for Special Research.

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