A decomposition theorem for singular K\"ahler spaces with trivial first Chern class of dimension at most four

TL;DR

This paper proves a Beauville-Bogomolov type decomposition for certain singular K"ahler fourfolds with trivial first Chern class, showing they decompose into products of tori, Calabi-Yau, and symplectic varieties, and explores related deformation and fundamental group properties.

Contribution

It establishes a decomposition theorem for singular K"ahler fourfolds with trivial first Chern class, extending classical results to singular settings and introducing a new approach to the Lipman-Zariski conjecture.

Findings

Existence of a Beauville-Bogomolov decomposition for the class of fourfolds considered.

Small projective deformations of these fourfolds are possible.

Fundamental group of such fourfolds is projective.

Abstract

Let $X$ be a compact K\"ahler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that $X$ admits a Beauville-Bogomolov decomposition: a finite quasi-\'etale cover of $X$ splits as a product of a complex torus and singular Calabi-Yau and irreducible holomorphic symplectic varieties. We also prove that $X$ has small projective deformations and the fundamental group of $X$ is projective. To obtain these results, we propose and study a new version of the Lipman-Zariski conjecture.