K\"ahler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

TL;DR

This paper studies compact K"ahler spaces with zero first Chern class, proving a Bochner principle for tensors, analyzing their structure via holonomy, and confirming Campana's Abelianity Conjecture in four dimensions.

Contribution

It establishes the Bochner principle for such spaces, classifies their tangent sheaves, and verifies Campana's Abelianity Conjecture in four-dimensional cases.

Findings

Holomorphic tensors are parallel with respect to singular Ricci-flat metrics.

Spaces split off a complex torus after a finite quasi-étale cover.

Classification of spaces with strongly stable tangent sheaf as Calabi–Yau or holomorphic symplectic.

Abstract

Let $X$ be a compact K\"ahler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of $X$: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-\'etale cover $X$ splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of $X$ according to its holonomy representation. In particular, we classify those $X$ which have strongly stable tangent sheaf: up to quasi-\'etale covers, these are either irreducible Calabi--Yau or irreducible holomorphic symplectic. As an application of these results, we show that if $X$ has dimension four, then it satisfies Campana's Abelianity Conjecture.