Torsion birational motives of surfaces and unramified cohomology

TL;DR

This paper explores the relationship between torsion birational motives of surfaces and unramified cohomology, demonstrating how trivial actions on cohomology influence other motivic invariants and providing a counterexample to the integral Hodge conjecture.

Contribution

It generalizes Kahn's result on torsion order of surfaces and links trivial cohomological actions to motivic functors, with an explicit example over complex numbers.

Findings

Trivial action on unramified cohomology implies trivial action on motivic functors.

Generalization of Kahn's torsion order result for surfaces.

Counterexample to the integral Hodge conjecture for a specific surface over .

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.