Streets-Tian Conjecture holds for 2-step solvmanifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913737180446720 |
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| author | Chen, Shuwen Zheng, Fangyang |
| author_facet | Chen, Shuwen Zheng, Fangyang |
| contents | A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be Kählerian (i.e., admitting a Kähler metric). The conjecture is known to be true in dimension $2$ but is still open in dimensions $3$ or higher. In this article, we confirm the conjecture for all 2-step solvmanifolds, namely, compact quotients of 2-step solvable Lie groups by discrete subgroups. In the proofs, we adopted a method of using special {\em non-unitary} frames, which enabled us to squeeze out some hidden symmetries to make the proof go through. Hopefully the technique could be further applied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10540 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Streets-Tian Conjecture holds for 2-step solvmanifolds Chen, Shuwen Zheng, Fangyang Differential Geometry 53C55 A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be Kählerian (i.e., admitting a Kähler metric). The conjecture is known to be true in dimension $2$ but is still open in dimensions $3$ or higher. In this article, we confirm the conjecture for all 2-step solvmanifolds, namely, compact quotients of 2-step solvable Lie groups by discrete subgroups. In the proofs, we adopted a method of using special {\em non-unitary} frames, which enabled us to squeeze out some hidden symmetries to make the proof go through. Hopefully the technique could be further applied. |
| title | Streets-Tian Conjecture holds for 2-step solvmanifolds |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2410.10540 |