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VOL. 70 | 2016 Singular spaces with trivial canonical class
Daniel Greb, Stefan Kebekus, Thomas Peternell

Editor(s) János Kollár, Osamu Fujino, Shigeru Mukai, Noboru Nakayama


The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau– and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation.

Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of projective varieties with canonical singularities and numerically trivial canonical class.

In view of recent progress in minimal model theory, this result can be seen as a first step towards a structure theory of manifolds with Kodaira dimension zero. Based on our main result, we argue that the natural building blocks for any structure theory are two classes of canonical varieties, which generalise the notions of irreducible Calabi-Yau– and irreducible holomorphic-symplectic manifolds, respectively.


Published: 1 January 2016
First available in Project Euclid: 4 October 2018

zbMATH: 1369.14052
MathSciNet: MR3617779

Digital Object Identifier: 10.2969/aspm/07010067

Primary: 14E30 , 14J32 , 32J27

Keywords: Beauville-Bogomolov Decomposition Theorem , Calabi-Yau , holomorphic-symplectic Varieties , minimal model theory , Varieties of Kodaira Dimension Zero

Rights: Copyright © 2016 Mathematical Society of Japan


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