Nilmanifolds with non-nilpotent complex structures and their pseudo-Kähler geometry
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arXiv
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| Format: | Preprint |
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2024
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| author | Latorre, A. Ugarte, L. |
| author_facet | Latorre, A. Ugarte, L. |
| contents | We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi-Yau structures. Moreover, we arrive at the topological restriction $b_1(X)\geq 3$ for every pseudo-Kähler nilmanifold $X$ with invariant complex structure, up to complex dimension four. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18692 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nilmanifolds with non-nilpotent complex structures and their pseudo-Kähler geometry Latorre, A. Ugarte, L. Differential Geometry Rings and Algebras 17B30, 53C15 (Primary) 32Q99, 53C50 (Secondary) We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi-Yau structures. Moreover, we arrive at the topological restriction $b_1(X)\geq 3$ for every pseudo-Kähler nilmanifold $X$ with invariant complex structure, up to complex dimension four. |
| title | Nilmanifolds with non-nilpotent complex structures and their pseudo-Kähler geometry |
| topic | Differential Geometry Rings and Algebras 17B30, 53C15 (Primary) 32Q99, 53C50 (Secondary) |
| url | https://arxiv.org/abs/2407.18692 |