K\'ahler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
| Autor: | Claudon, Benoît, Graf, Patrick, Guenancia, Henri, Naumann, Philipp |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1515/crelle-2022-0001 |
| Popis: | Let $X$ be a compact K\"ahler 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-\'etale 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 stable tangent sheaf: up to quasi-\'etale covers, these are either irreducible Calabi--Yau or irreducible holomorphic symplectic. As an application of these results, we show that if $X$ has dimension four, then it satisfies Campana's Abelianity Conjecture. Comment: 28 pages; v2: exposition improved following the referee's suggestions, new title, section 7 shortened and section 8 removed; to appear in Crelle |
| Databáze: | arXiv |
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