Exotic hypercomplex structures on a torus do not exist
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909656560959488 |
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| author | Federico, Alberto Pipitone Verbitsky, Misha |
| author_facet | Federico, Alberto Pipitone Verbitsky, Misha |
| contents | A hypercomplex manifold is a manifold with three complex structures satisfying quaternionic relations. Such a manifold admits a unique torsion-free connection preserving the quaternionic action, called the Obata connection. A compact Kahler manifold admitting a hypercomplex structure always admits a hyperkahler structure as well; however, it is not obvious whether the original hypercomplex structure is hyperkahler. A non-hyperkahler hypercomplex structure on a Kahler manifold is called exotic. We show that the Obata connection for an exotic hypercomplex structure on a torus is flat and classify complete flat affine structures on real tori. We use this classification to prove that exotic hypercomplex structures do not exist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18179 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exotic hypercomplex structures on a torus do not exist Federico, Alberto Pipitone Verbitsky, Misha Differential Geometry Algebraic Geometry 53C26, 53C28, 53C55 A hypercomplex manifold is a manifold with three complex structures satisfying quaternionic relations. Such a manifold admits a unique torsion-free connection preserving the quaternionic action, called the Obata connection. A compact Kahler manifold admitting a hypercomplex structure always admits a hyperkahler structure as well; however, it is not obvious whether the original hypercomplex structure is hyperkahler. A non-hyperkahler hypercomplex structure on a Kahler manifold is called exotic. We show that the Obata connection for an exotic hypercomplex structure on a torus is flat and classify complete flat affine structures on real tori. We use this classification to prove that exotic hypercomplex structures do not exist. |
| title | Exotic hypercomplex structures on a torus do not exist |
| topic | Differential Geometry Algebraic Geometry 53C26, 53C28, 53C55 |
| url | https://arxiv.org/abs/2506.18179 |