A strong counterexample to the log canonical Beauville--Bogomolov decomposition

TL;DR

The paper constructs a counterexample in algebraic geometry showing that the Beauville--Bogomolov decomposition does not hold in the log canonical setting, and explores related properties of Albanese morphisms.

Contribution

It provides the first strong counterexample to the log canonical Beauville--Bogomolov decomposition and studies the stability of Albanese morphisms for certain pairs.

Findings

Constructs a d-dimensional log canonical, K-trivial variety with non-birational general fibers.

Shows the existence of a smooth quasi-projective variety with maximal variation in its quasi-Albanese morphism.

Proves that the Albanese morphism for log canonical pairs with nef anti-canonical class is locally stable.

Abstract

For every $d \geq 4$, we construct a $d$-dimensional, log canonical, $K$-trivial variety with the property that two general fibers of its Albanese morphism are not birational. This provides a strong counterexample to the Beauville--Bogomolov decomposition in the log canonical setting. This construction can also be adapted to construct a smooth quasi-projective variety of logarithmic Kodaira dimension 0 whose quasi-Albanese morphism has maximal variation. On the positive side, we show that the Albanese morphism for log canonical pairs with nef anti-canonical class is a locally stable family of pairs.