Singular spaces with trivial canonical class

Abstract

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.

Algebraic integrability of foliations with numerically trivial canonical bundle

“…Proof of the claim. Since F is strongly stable in the sense of [GKP16c,Defn.7 Arguing by contradiction suppose that T X is pseudoeffective. Hence by Theorem 1.1 we have c 2 (X) · H n−2 = 0.…”

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

“…corresponds to the decomposition of T x X into irreducible representations under the action of the restricted holonomy group Hol(X reg , g H ) at x. Proposition D is a significant refinement of the decompositions obtained in [GKP16b] and [Gue16] and, as already mentioned above, is one of the ingredients in Höring-Peternell's proof for the singular analogue of the Decomposition Theorem in any dimension, see [HP17]. The irreducible pieces appearing in their result are the ones described in [GKP16b, Sect.…”

“…Motivated by a decomposition theorem for tangent sheaves of klt varieties with numerically trivial canonical divisor established by Greb, Kebekus, and Peternell in [GKP16b] and building on Bost's criteria for algebraic integrability of foliations, Druel recently obtained the singular version of the Decomposition Theorem for such varieties of dimension dim X ≤ 5, see [Dru18]. Using Druel's strategy as well as the results presented in this paper, in particular Proposition D and Theorem E below, Höring and Peternell very recently gave a proof of the singular version of the Decomposition Theorem in [HP17], thus completing the long quest for such a result in the singular category.…”

“…Decomposition of the tangent sheaf. Using the holonomy principle as well as the classification of Ricci-flat restricted holonomy groups, and comparing with the decomposition of the tangent sheaf of varieties with trivial canonical divisor established in [GKP16b], we can obtain more precise information about the tangent sheaf of Z, which splits according to the restricted holonomy representation of X.…”

Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class

“…8.2], normal projective varieties with only canonical singularities and K X numerically equivalent to zero have κ(X) = 0. In the following statement of a theorem of Kawamata, we have also incorporated some remarks of [27]. ) Let X be a normal connected projective variety with at worst canonical singularities and K X numerically equivalent to zero.…”