Singular spaces with trivial canonical class

By D. Greb, Stefan Kebekus, T. Peternell. 2011

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The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact K\"ahler manifold with numerically trivial canonical bundle admits an \'etale 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6eb6ed8b0860e0651f36a0490cc4a0666beba66c

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Citations in the corpus

Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups (2017, methodology): “-ﬂat restricted holonomy groups, and comparing with the decomposition of the tangent sheaf of varieties with trivial canonical divisor 4 DANIEL GREB, HENRI GUENANCIA, AND STEFANKEBEKUS established in [GKP16b], we can obtain more precise information about the tangent sheaf of Z, which splits accordin”