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

Daniel Greb (1) , Henri Guenancia (2, 3) , Stefan Kebekus (4)

1 Universität Duisburg-Essen = University of Duisburg-Essen [Essen]
2 CNRS - Centre National de la Recherche Scientifique
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 Mathematisches Institut [Freiburg]

We investigate the holonomy group of singular K\"ahler-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-\'etale covers, varieties with strongly stable tangent sheaf are either Calabi-Yau or irreducible holomorphic symplectic. These results form one building block for H\"oring-Peternell's recent proof of a singular version of the Beauville-Bogomolov Decomposition Theorem.

Domains: Algebraic Geometry [math.AG]; Differential Geometry [math.DG]

Daniel Greb, Henri Guenancia, Stefan Kebekus. Klt varieties with trivial canonical class - Holonomy, differential forms, and fundamental groups. Geometry and Topology, 2019, ⟨10.2140/gt.2019.23.2051⟩. ⟨hal-01879010⟩