Klt varieties with trivial canonical class -- Holonomy, differential forms, and fundamental groups

Abstract

We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tangent sheaf are established. As a consequence, known decompositions for tangent sheaves of varieties with trivial canonical divisor are refined. In particular, we show that up to finite quasi-étale covers, varieties with strongly stable tangent sheaf are either Calabi-Yau or irreducible holomorphic symplectic. These results form one building block for Höring-Peternell's recent proof of a singular version of the Beauville-Bogomolov Decomposition Theorem.

v4: final version, to appear in Geometry & Topology; v5: filled small gap in the proof of Cor. 7.4

Cited by in corpus (13)

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• Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
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• Two Non-Vanishing results concerning the Anti-Canonical Bundle
• A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four
• Fundamental groups of log Calabi-Yau surfaces