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

Daniel Greb, Henri Guenancia and Stefan Kebekus

Geometry & Topology 23 (2019) 2051–2124
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 and Peternell’s recent proof of a singular version of the Beauville–Bogomolov decomposition theorem.

Keywords
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
Mathematical Subject Classification 2010
Primary: 14E30, 14J32, 32J27
References
Publication
Received: 6 November 2017
Accepted: 2 December 2018
Published: 17 June 2019
Proposed: Dan Abramovich
Seconded: Richard Thomas, Jim Bryan
Authors
Daniel Greb
Essener Seminar für Algebraische Geometrie und Arithmetik
Fakultät für Mathematik
Universität Duisburg–Essen
Essen
Germany
http://www.esaga.uni-due.de/daniel.greb
Henri Guenancia
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States
Institut de Mathématiques de Toulouse
Université Paul Sabatier
Toulouse
France
https://www.math.univ-toulouse.fr/~hguenanc/
Stefan Kebekus
Mathematisches Institut
Albert-Ludwigs-Universität Freiburg
Freiburg im Breisgau
Germany
Freiburg Institute for Advanced Studies
Freiburg im Breisgau
Germany
University of Strasbourg Institute for Advanced Study
Strasbourg
France
https://cplx.vm.uni-freiburg.de