Advanced Studies in Pure Mathematics
- Adv. Stud. Pure Math.
- Minimal Models and Extremal Rays (Kyoto, 2011), J. Kollár, O. Fujino, S. Mukai and N. Nakayama, eds. (Tokyo: Mathematical Society of Japan, 2016), 67 - 113
Singular spaces with trivial canonical class
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.
Received: 12 December 2011
Revised: 29 July 2012
First available in Project Euclid: 4 October 2018
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Greb, Daniel; Kebekus, Stefan; Peternell, Thomas. Singular spaces with trivial canonical class. Minimal Models and Extremal Rays (Kyoto, 2011), 67--113, Mathematical Society of Japan, Tokyo, Japan, 2016. doi:10.2969/aspm/07010067. https://projecteuclid.org/euclid.aspm/1538622705