A decomposition theorem for singular spaces with trivial canonical class of dimension at most five

TL;DR

This paper extends the Beauville-Bogomolov decomposition theorem to certain singular complex projective varieties of dimension up to five, showing they can be decomposed into products involving Abelian varieties and singular analogues of Calabi-Yau and holomorphic symplectic varieties.

Contribution

It provides a partial extension of the decomposition theorem to singular varieties with trivial canonical class in dimensions up to five.

Findings

Varieties admit a finite cover decomposing into known types.

Decomposition involves Abelian, Calabi-Yau, and holomorphic symplectic components.

Applicable to varieties with canonical singularities and trivial canonical class.

Abstract

In this paper we partly extend the Beauville-Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, \'etale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible holomorphic symplectic varieties.