Regular homomorphisms, with a twist

TL;DR

This paper introduces a new perspective on regular homomorphisms by associating torsors over an abelian variety to certain cycle data, providing obstructions to algebraic cycle existence and linking to rationality problems of threefolds.

Contribution

It demonstrates that cycle data without $K$-points determines torsors over $K$, offering a novel obstruction to algebraic cycle existence and connecting to recent rationality results.

Findings

Cycle data without $K$-points defines torsors over $K$.

Obstruction to algebraic cycle existence over $K$.

Links to rationality of threefolds via recent results.

Abstract

Let $X/K$ be a variety over a field, and $A/K$ an abelian variety. A regular homomorphism to $A$ (in codimension $i$) induces, for every smooth geometrically connected pointed $K$-scheme $(T,t_0)$ and every cycle class $Z \in CH^i(T\times X)$, a morphism $T \to A$ of varieties over $K$. In this note we show that, if $T$ admits no $K$-point, the data $(T,Z)$ determines a torsor $A^{(T,Z)}$ over $K$ under $A$ and a $K$-morphism $T \to A^{(T,Z)}$. This can be used to provide an obstruction to the existence of algebraic cycles defined over $K$. We then connect this obstruction to some recent results of Hassett--Tschinkel and Benoist--Wittenberg on rationality of threefolds.