2022
DOI: 10.1515/crelle-2022-0001
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Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

Abstract: Let X be a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of X: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover X splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of X according to its holonomy representation. In particular, we classify those X which have strongly stabl… Show more

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Cited by 7 publications
(19 citation statements)
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“…Even more recently, Chen and Wentworth [9] have likewise obtained a Bogomolov-Gieseker inequality in a setting similar to ours. Their results, once combined with our previous paper [10], readily imply that if X is a compact Kähler space with klt singularities, smooth in codimension two and c 1 .X/ D 0, then there exists an admissible Hermite-Einstein metric h on T X reg . Assuming additionally that c 2 .X/ ˛n 2 D 0…”
Section: Introductionmentioning
confidence: 52%
See 2 more Smart Citations
“…Even more recently, Chen and Wentworth [9] have likewise obtained a Bogomolov-Gieseker inequality in a setting similar to ours. Their results, once combined with our previous paper [10], readily imply that if X is a compact Kähler space with klt singularities, smooth in codimension two and c 1 .X/ D 0, then there exists an admissible Hermite-Einstein metric h on T X reg . Assuming additionally that c 2 .X/ ˛n 2 D 0…”
Section: Introductionmentioning
confidence: 52%
“…Also, p a is a Kähler class by [16,Proposition 3.6]. Finally, if the conclusion of Theorem 5.2 holds for z X, then it also holds for X , by taking Galois closure [10,Lemma 2.8]. We may and will therefore replace X by z X (and a by p a) for the remaining argument.…”
Section: Characterization Of Torus Quotientsmentioning
confidence: 99%
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“…We claim that Z is not a torus. Otherwise, by Lemma 2.8 of [CGGN22], let Z ′ → Z → Y be the Galois cover of Z → Y . By construction, Z ′ → Z is quasi-étale.…”
Section: [P]mentioning
confidence: 99%
“…where each F min i is α-stable with zero first Chern class and its restriction to the regular locus of X min is a parallel subbundle with respect to ω min (see [GGK, Proposition D] and [CGGN,Remark 3.5]). Using φ, we get a similar holomorphic decomposition…”
Section: Moishezon Manifolds With Vanishing First Chern Classmentioning
confidence: 99%