Geometry & Topology

Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups

Daniel Greb, Henri Guenancia, and Stefan Kebekus

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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 and Peternell’s recent proof of a singular version of the Beauville–Bogomolov decomposition theorem.

Article information

Geom. Topol., Volume 23, Number 4 (2019), 2051-2124.

Received: 6 November 2017
Accepted: 2 December 2018
First available in Project Euclid: 16 July 2019

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Zentralblatt MATH identifier

Primary: 14E30: Minimal model program (Mori theory, extremal rays) 14J32: Calabi-Yau manifolds 32J27: Compact Kähler manifolds: generalizations, classification

varieties with trivial canonical divisor klt singularities Kähler–Einstein metrics stability holonomy groups Bochner principle irreducible holomorphic symplectic varieties Calabi–Yau varieties differential forms fundamental groups decomposition


Greb, Daniel; Guenancia, Henri; Kebekus, Stefan. Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups. Geom. Topol. 23 (2019), no. 4, 2051--2124. doi:10.2140/gt.2019.23.2051.

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