On the cancellation problem for algebraic tori

TL;DR

This paper investigates the cancellation problem for algebraic tori, demonstrating that the property fails in all dimensions greater than or equal to two, contrasting with known positive cases in lower dimensions.

Contribution

It proves that the cancellation property does not hold universally for varieties of dimension two or higher, providing a counterexample to a longstanding question.

Findings

Cancellation holds for curves and certain smooth varieties.

Counterexamples show cancellation fails in dimensions ≥ 2.

The result clarifies limitations of the cancellation property in algebraic geometry.

Abstract

We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for curves, is known to hold for smooth varieties of log-general type by virtue of a result of Iitaka-Fujita, and more generally for varieties which are not dominantly covered by images of the punctured affine line. We show in contrast that cancellation fails in general in every dimension bigger or equal to two.