A note on almost abelian groups with constant holomorphic sectional curvature
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914396676030464 |
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| author | Li, Yulu Zheng, Fangyang |
| author_facet | Li, Yulu Zheng, Fangyang |
| contents | A long-standing conjecture in non-Kähler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant $c$, then the metric must be Kähler when $c\neq 0$ and must be Chern (or Levi-Civita) flat when $c=0$. The conjecture is known to be true in dimension 2 by the work of Balas-Gauduchon, Sato-Sekigawa, and Apostolov-Davidov-Muskarov in the 1980s and 1990s. In dimension 3 or higher, the conjecture is still open except in some special cases, such as for all twistor spaces by Davidov-Grantcharov-Muskarov, for locally conformally Kähler manifolds (when $c\leq 0$) by Chen-Chen-Nie, etc. In this short note, we consider compact quotients $G/Γ$ where $G$ is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and $Γ$ is a discrete subgroup. We confirm the conjecture when the Lie algebra ${\mathfrak g}$ of $G$ either is almost abelian, or contains a $J$-invariant abelian ideal of codimension 2. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_00415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on almost abelian groups with constant holomorphic sectional curvature Li, Yulu Zheng, Fangyang Differential Geometry 53C55 A long-standing conjecture in non-Kähler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant $c$, then the metric must be Kähler when $c\neq 0$ and must be Chern (or Levi-Civita) flat when $c=0$. The conjecture is known to be true in dimension 2 by the work of Balas-Gauduchon, Sato-Sekigawa, and Apostolov-Davidov-Muskarov in the 1980s and 1990s. In dimension 3 or higher, the conjecture is still open except in some special cases, such as for all twistor spaces by Davidov-Grantcharov-Muskarov, for locally conformally Kähler manifolds (when $c\leq 0$) by Chen-Chen-Nie, etc. In this short note, we consider compact quotients $G/Γ$ where $G$ is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and $Γ$ is a discrete subgroup. We confirm the conjecture when the Lie algebra ${\mathfrak g}$ of $G$ either is almost abelian, or contains a $J$-invariant abelian ideal of codimension 2. |
| title | A note on almost abelian groups with constant holomorphic sectional curvature |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2503.00415 |