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.

arXiv commentv4: final version, to appear in Geometry & Topology; v5: filled small gap in the proof of Cor. 7.4

References in corpus (7)

• Differential Forms on Log Canonical Spaces
• Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties
• Singular symplectic moduli spaces
• Symmetric differentials and the fundamental group
• Movable curves and semistable sheaves
• Reflexive differential forms on singular spaces -- Geometry and Cohomology
• Calabi-Yau quotients with terminal singularities

Cited by in corpus (15)

• The global moduli theory of symplectic varieties
• Codimension one foliations with numerically trivial canonical class on singular spaces
• Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities
• Birational boundedness of rationally connected Calabi-Yau 3-folds
• Orbifold Stability and Miyaoka-Yau Inequality for minimal pairs
• The integral Hodge conjecture for 3-folds of Kodaira dimension zero
• Fujiki relations and fibrations of irreducible symplectic varieties
• The Nonvanishing problem for varieties with nef anticanonical bundle
• Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
• On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four
• Free abelian group actions on normal projective varieties: sub-maximal dynamical rank case
• Semispecial tensors and quotients of the polydisc
• Two Non-Vanishing results concerning the Anti-Canonical Bundle
• Fundamental groups of log Calabi-Yau surfaces
• A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four