On the existence of balanced metrics of Hodge-Riemann type
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915769005113344 |
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| author | Fino, Anna Mainenti, Asia |
| author_facet | Fino, Anna Mainenti, Asia |
| contents | In the paper we study the existence of balanced metrics of Hodge-Riemann type on non-Kähler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be $(n - 2)$-Kähler. Then, we focus on the case of compact quotients of Lie groups by lattices, endowed with an invariant complex structure. In particular, we prove non existence results on non-Kähler complex parallelizable manifolds and some classes of solvmanifolds, and we show that the only nilmanifolds admitting invariant structures of this type are tori. Finally, we construct the first non-Kähler example of a Hodge-Riemann balanced structure, on a non-compact complex manifold obtained as the product of the Iwasawa manifold by $\mathbb C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of balanced metrics of Hodge-Riemann type Fino, Anna Mainenti, Asia Differential Geometry Complex Variables In the paper we study the existence of balanced metrics of Hodge-Riemann type on non-Kähler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be $(n - 2)$-Kähler. Then, we focus on the case of compact quotients of Lie groups by lattices, endowed with an invariant complex structure. In particular, we prove non existence results on non-Kähler complex parallelizable manifolds and some classes of solvmanifolds, and we show that the only nilmanifolds admitting invariant structures of this type are tori. Finally, we construct the first non-Kähler example of a Hodge-Riemann balanced structure, on a non-compact complex manifold obtained as the product of the Iwasawa manifold by $\mathbb C$. |
| title | On the existence of balanced metrics of Hodge-Riemann type |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2408.14377 |