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

Greb, Daniel¹ ; Guenancia, Henri² ; Kebekus, Stefan³

¹ Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg–Essen, Essen, Germany
² Department of Mathematics, Stony Brook University, Stony Brook, NY, United States, Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France
³ 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

Geometry & topology, Tome 23 (2019) no. 4, pp. 2051-2124

Résumé

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.

Détail

DOI : 10.2140/gt.2019.23.2051

Classification : 14E30, 14J32, 32J27
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

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