Verbal ideals and unobstructed complex parallelisable nilmanifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929605933268992 |
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| author | Paulsen, Matthias Rollenske, Sönke Wehler, Konstantin |
| author_facet | Paulsen, Matthias Rollenske, Sönke Wehler, Konstantin |
| contents | We show that a compact complex parallelisable nilmanifold has unobstructed deformations if and only if its associated Lie algebra satisfies a reality condition and is a free Lie algebra in a variety of Lie algebras, that is, defined by a verbal ideal in a free Lie algebra. We provide a partial classification of verbal ideals and show that there are finitely many such Lie algebras up to dimension 19, whereas infinite families start to appear in dimension 20. As a consequence, there are finitely many complex homotopy types of unobstructed complex parallelisable nilmanifolds up to dimension 19, and infinitely many in dimension 20. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Verbal ideals and unobstructed complex parallelisable nilmanifolds Paulsen, Matthias Rollenske, Sönke Wehler, Konstantin Differential Geometry Algebraic Geometry Rings and Algebras 32G05 (Primary) 17B01, 17B30, 32M10 (Secondary) We show that a compact complex parallelisable nilmanifold has unobstructed deformations if and only if its associated Lie algebra satisfies a reality condition and is a free Lie algebra in a variety of Lie algebras, that is, defined by a verbal ideal in a free Lie algebra. We provide a partial classification of verbal ideals and show that there are finitely many such Lie algebras up to dimension 19, whereas infinite families start to appear in dimension 20. As a consequence, there are finitely many complex homotopy types of unobstructed complex parallelisable nilmanifolds up to dimension 19, and infinitely many in dimension 20. |
| title | Verbal ideals and unobstructed complex parallelisable nilmanifolds |
| topic | Differential Geometry Algebraic Geometry Rings and Algebras 32G05 (Primary) 17B01, 17B30, 32M10 (Secondary) |
| url | https://arxiv.org/abs/2411.17560 |