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A decomposition theorem for singular spaces with trivial canonical class of dimension at most five. (English) Zbl 1419.14063
Let \(X\) be a compact Kähler manifold with numerically trivial canonical bundle. A very important Beauville-Bogomolov decomposition theorem says that such varieties admit an étale cover that decomposes into a product of a torus, irreducible simply connected Calabi-Yau and holomorphic symplectic manifolds.
The main aim of the paper is to generalize this theorem to singular varieties appearing in the minimal model program. This is achieved only partially for normal projective varieties of dimension at most \(5\) with klt singularities. The smooth case uses Kähler-Einstein metrics and Yau’s proof of the Calabi conjecture. In the singular setting the method of proof is different and it uses a generalization of J.-B. Bost’s criterion [Publ. Math., Inst. Hautes tud. Sci. 93, 161–221 (2001; Zbl 1034.14010)] on algebraicity of leaves of a foliation. This result is combined with an earlier work of D. Greb et al. [Adv. Stud. Pure Math. 70, 67–113 (2016; Zbl 1369.14052)] that describes a weak version of the Beauville-Bogomolov decomposition.
An important role in the proof plays Theorem 6.1 describing rank at most \(3\) stable reflexive sheaves with zero first Chern class. This result was the main obtsacle in generalizing the results to higher dimensions. Recently, A. Höring and T. Peternell [Invent. Math. 216, No. 2, 395–419 (2019; Zbl 07061101)] managed to generalize this theorem to higher rank. This result, together with Druels’s ideas, allows to prove a general version of the Beauville-Bogomolov decomposition in the singular case.
Reviewer’s remark: The proof of Proposition 5.4 is incomplete. However, in the proof of the main results one needs only a weaker version where the proof works (see the author’s erratum on http://druel.perso.math.cnrs.fr/publications.html).

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
37F75 Dynamical aspects of holomorphic foliations and vector fields
14E30 Minimal model program (Mori theory, extremal rays)
32C99 Analytic spaces
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