\(\mathrm{klt}\) varieties with trivial canonical class: holonomy, differential forms, and fundamental groups.

*(English)*Zbl 1423.14110Let \(X\) be a compact complex variety in the Fujiki class \(\mathscr{C}\) with mild singularities (e.g. with klt singularities). It is an important task in the study of bimeromorphic classification of Kähler manifolds to comprehend the structure of such \(X\) since they appear naturally as the minimal models of compact Kähler manifolds of Kodaira dimension \(0\).

If \(X\) is smooth, it is well known, by the Beauville-Bogomomlov Decomposition Theorem (see e.g. A. Beauville [J. Differ. Geom. 18, 755–782 (1983; Zbl 0537.53056)]), that \(X\) admits a finite étale cover which decomposes as a product of a complex torus, irreducible simply connected Calabi-Yau manifolds and holomorphic symplectic manifolds. The article under review is among the works that attempt to generalize the Beauviell-Bogomolov decomposition to the singular case, a study initiated in [D. Greb et al., Adv. Stud. Pure Math. 70, 67–113 (2016; Zbl 1369.14052)] where they also propose a definition of singular counterparts of Calabi-Yau/holomorphic symplectic manifolds.

Quite different from the classical approach to prove the Beauville-Bogomolov decomposition, in the singular case the main idea is to find a natural splitting of the tangent sheaf of \(X\) into algebraically integrable foliations and then prove that this splitting gives rise to a decomposition of \(X\) up to a quasi-étale cover. In [D. Greb et al., Adv. Stud. Pure Math. 70, 67–113 (2016; Zbl 1369.14052)], (up to a quasi-étale cover) a decomposition of the tangent sheaf of \(X\) into strongly stable foliations with trivial determinants is obtained for \(X\) klt projective; and by the further works of S. Druel [Invent. Math. 211, No. 1, 245–296 (2018; Zbl 1419.14063)] and A. Höring and Th. Peternell [Invent. Math. 216, No. 2, 395–419 (2019; Zbl 07061101)], the direct summands in the decomposition are algebraically integrable and give rise to a decomposition of \(X\) up to finite quasi-étale cover.

In order to obtain the singular version of the Beauville-Bogomolov decomosition for klt projective varieties, it remains to show that the non-flat direct summands in the decomposition of the tangent sheaf correponds to singular irreducible Calabi-Yau varieties and singular irreducible holomorphic symplectic varieties. This is achieved in the article under review in three steps: First, by Ph. Eyssidieux et al. [J. Am. Math. Soc. 22, No. 3, 607–639 (2009; Zbl 1215.32017)], (with respect to a fixed polarization \(H\)) one has a canonical positive current \(\omega_H\) with bounded potentials whose restrcition to \(X_{\text{reg}}\) is a genuine Ricci-flat metric, then one can use this metric to define the holonomy groups at smooth points of \(X\), which induce a decomposition of the tangent sheaf (the “canonical decomposition”) according to the irreducible decomposition of the holonomy action. It is then easy to show that the summands are strongly stable and integrable (Proposition 6.6, (6.3.2)) by D.Greb et al. [Adv. Stud. Pure Math. 70, 67–113 (2016; Zbl 1369.14052)]. Second, by passing to the “holonomy cover” (Theorem 7.1) which renders the holonomy group connected, the authors prove in Proposition D (Proposition 7.9) that the holonomy according to any non-flat summand is either \(\text{SU}(m)\) or \(\text{Sp}(m)\). In the construction of “holonomy cover”, a result of S. Druel [Invent. Math. 211, No. 1, 245–296 (2018; Zbl 1419.14063)] is used to control the size of the flat summand. Third, the authors give a characterization of singular irreducible Calabi-Yau varieties/holomorphic symplectic varieties in terms of the holonomy. This is achieved in Proposition E (Proposition 12.10) by applying the Bochner principle (Theorem 8.1 & Theorem A), which states that the (reflexive) holomorphic tensors are parallel with respect to \(\omega_H\). The Bochner principle (Theorem 8.1) is obtained by an argument similar to that in [H. Guenancia, Algebr. Geom. 3, No. 5, 508–542 (2016; Zbl 1379.32010)]: estimate the slope and the second fundamental form by constructing a sequence of smooth metrics regularizing the pull-back of \(\omega_H\) to a (log) desingularization of \(X\).

In the last part of the article the authors also compare their definition of singular Calabi-Yau/holomorphic symplectic varieties in terms of holonomy with the ones in the litteratures (§14). Especially they point out by an example (Example 14.9) that, even when \(X\) is Gorenstein and has trivial canonical bundle, one cannot replace “holonomy” by “restricted holonomy” in the definition (while it is the case in the smooth setting).

If \(X\) is smooth, it is well known, by the Beauville-Bogomomlov Decomposition Theorem (see e.g. A. Beauville [J. Differ. Geom. 18, 755–782 (1983; Zbl 0537.53056)]), that \(X\) admits a finite étale cover which decomposes as a product of a complex torus, irreducible simply connected Calabi-Yau manifolds and holomorphic symplectic manifolds. The article under review is among the works that attempt to generalize the Beauviell-Bogomolov decomposition to the singular case, a study initiated in [D. Greb et al., Adv. Stud. Pure Math. 70, 67–113 (2016; Zbl 1369.14052)] where they also propose a definition of singular counterparts of Calabi-Yau/holomorphic symplectic manifolds.

Quite different from the classical approach to prove the Beauville-Bogomolov decomposition, in the singular case the main idea is to find a natural splitting of the tangent sheaf of \(X\) into algebraically integrable foliations and then prove that this splitting gives rise to a decomposition of \(X\) up to a quasi-étale cover. In [D. Greb et al., Adv. Stud. Pure Math. 70, 67–113 (2016; Zbl 1369.14052)], (up to a quasi-étale cover) a decomposition of the tangent sheaf of \(X\) into strongly stable foliations with trivial determinants is obtained for \(X\) klt projective; and by the further works of S. Druel [Invent. Math. 211, No. 1, 245–296 (2018; Zbl 1419.14063)] and A. Höring and Th. Peternell [Invent. Math. 216, No. 2, 395–419 (2019; Zbl 07061101)], the direct summands in the decomposition are algebraically integrable and give rise to a decomposition of \(X\) up to finite quasi-étale cover.

In order to obtain the singular version of the Beauville-Bogomolov decomosition for klt projective varieties, it remains to show that the non-flat direct summands in the decomposition of the tangent sheaf correponds to singular irreducible Calabi-Yau varieties and singular irreducible holomorphic symplectic varieties. This is achieved in the article under review in three steps: First, by Ph. Eyssidieux et al. [J. Am. Math. Soc. 22, No. 3, 607–639 (2009; Zbl 1215.32017)], (with respect to a fixed polarization \(H\)) one has a canonical positive current \(\omega_H\) with bounded potentials whose restrcition to \(X_{\text{reg}}\) is a genuine Ricci-flat metric, then one can use this metric to define the holonomy groups at smooth points of \(X\), which induce a decomposition of the tangent sheaf (the “canonical decomposition”) according to the irreducible decomposition of the holonomy action. It is then easy to show that the summands are strongly stable and integrable (Proposition 6.6, (6.3.2)) by D.Greb et al. [Adv. Stud. Pure Math. 70, 67–113 (2016; Zbl 1369.14052)]. Second, by passing to the “holonomy cover” (Theorem 7.1) which renders the holonomy group connected, the authors prove in Proposition D (Proposition 7.9) that the holonomy according to any non-flat summand is either \(\text{SU}(m)\) or \(\text{Sp}(m)\). In the construction of “holonomy cover”, a result of S. Druel [Invent. Math. 211, No. 1, 245–296 (2018; Zbl 1419.14063)] is used to control the size of the flat summand. Third, the authors give a characterization of singular irreducible Calabi-Yau varieties/holomorphic symplectic varieties in terms of the holonomy. This is achieved in Proposition E (Proposition 12.10) by applying the Bochner principle (Theorem 8.1 & Theorem A), which states that the (reflexive) holomorphic tensors are parallel with respect to \(\omega_H\). The Bochner principle (Theorem 8.1) is obtained by an argument similar to that in [H. Guenancia, Algebr. Geom. 3, No. 5, 508–542 (2016; Zbl 1379.32010)]: estimate the slope and the second fundamental form by constructing a sequence of smooth metrics regularizing the pull-back of \(\omega_H\) to a (log) desingularization of \(X\).

In the last part of the article the authors also compare their definition of singular Calabi-Yau/holomorphic symplectic varieties in terms of holonomy with the ones in the litteratures (§14). Especially they point out by an example (Example 14.9) that, even when \(X\) is Gorenstein and has trivial canonical bundle, one cannot replace “holonomy” by “restricted holonomy” in the definition (while it is the case in the smooth setting).

Reviewer: Juanyong Wang (Palaiseau)

##### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

32J27 | Compact Kähler manifolds: generalizations, classification |

##### Keywords:

stability; fundamental groups; differential forms; Kähler-Einstein metrics; Calabi-Yau varieties; Bochner principle; irreducible holomorphic symplectic varieties; singular Beauville-Bogmolov decomposition
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\textit{D. Greb} et al., Geom. Topol. 23, No. 4, 2051--2124 (2019; Zbl 1423.14110)

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