Torsion birational motives of surfaces and unramified cohomology

Author:

Kanetomo Sato, Takao Yamazaki

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), K-Theory and Homology (math.KT), Number Theory (math.NT)

journal:

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date:

2023-08-31 16:00:00

Abstract

Let $S$ and $T$ be smooth projective varieties over an algebraically closed field. 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.