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

## TL;DR

This paper studies the holonomy groups and geometric structures of singular Kähler-Einstein varieties with trivial canonical class, refining their classification and contributing to the singular Beauville-Bogomolov decomposition.

## Contribution

It establishes new links between holonomy irreducibility, stability of tangent sheaves, and the structure of varieties with trivial canonical class, advancing the understanding of their geometric decomposition.

## Key findings

- Finiteness of connected components of holonomy groups
- A Bochner principle for holomorphic tensors on these varieties
- Varieties with stable tangent sheaves are either Calabi-Yau or irreducible holomorphic symplectic

## Abstract

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.