On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action

TL;DR

This paper proves that under certain symmetry conditions, the Albanese morphism of a log Calabi-Yau pair is birationally isotrivial, extending understanding of the structure of such pairs and avoiding known counterexamples.

Contribution

It establishes birational isotriviality of the Albanese morphism for log Calabi-Yau pairs with torus symmetries, providing conditions to exclude pathological cases.

Findings

Two general fibers of the Albanese morphism are birationally equivalent.

The existence of a large enough torus in automorphisms prevents non-isotrivial examples.

The result applies to pairs with specific symmetry properties, refining previous counterexamples.

Abstract

Let $(X,\Delta)$ be a projective, log canonical, $K$-trivial pair over the complex numbers. Let $Z$ be a minimal log canonical center of $(X,\Delta)$ and suppose that there exists a torus $\mathbb{T}\subseteq\operatorname{Aut}(X)$ preserving $\Delta$ and such that $\dim\mathbb{T}=\operatorname{codim}_X Z$. Then we show that two general fibers of the Albanese morphism $\operatorname{alb}_X$ are birationally equivalent. In particular, the pathological example of a projective, log canonical, $K$-trivial variety whose Albanese morphism is not generically birationally isotrivial, recently constructed by Bernasconi, Filipazzi, Patakfalvi and Tsakanikas, can be avoided under the additional hypothesis that there exists a torus of large enough dimension in the automorphism group of the given pair.