Non-Kähler metrics on complex manifolds of LVMB type
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916056218468352 |
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| author | Thiella, Federico |
| author_facet | Thiella, Federico |
| contents | LVMB manifolds are a class of non-Kähler compact complex manifolds with a remarkably rich geometry: in many cases they admit a holomorphic bundle structure over a compact toric manifold. In fact, such a bundle is determined by an algebro-combinatoric datum encapsulating a simplicial fan. This is reflected in a close relationship between the geometry of an LVMB manifold and that of its toric base space. Throughout this paper, we restrict to this subclass of LVMB manifolds. We provide a formula for the characteristic class of the bundle in terms of the original LVMB datum. Subsequently, this expression is employed to address the existence of Hermitian metrics satisfying ``special'' conditions. We consider balanced and SKT metrics, showing that the former are obstructed in most cases. Moreover, within the LVMB class, we construct a new example of a manifold admitting a balanced metric. Finally, restricting ourselves to the subclass of LVM manifolds, we characterize those admitting an SKT metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28621 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Kähler metrics on complex manifolds of LVMB type Thiella, Federico Differential Geometry Complex Variables 32Q99, 53C55, 14M25, 32L05, 32M25 LVMB manifolds are a class of non-Kähler compact complex manifolds with a remarkably rich geometry: in many cases they admit a holomorphic bundle structure over a compact toric manifold. In fact, such a bundle is determined by an algebro-combinatoric datum encapsulating a simplicial fan. This is reflected in a close relationship between the geometry of an LVMB manifold and that of its toric base space. Throughout this paper, we restrict to this subclass of LVMB manifolds. We provide a formula for the characteristic class of the bundle in terms of the original LVMB datum. Subsequently, this expression is employed to address the existence of Hermitian metrics satisfying ``special'' conditions. We consider balanced and SKT metrics, showing that the former are obstructed in most cases. Moreover, within the LVMB class, we construct a new example of a manifold admitting a balanced metric. Finally, restricting ourselves to the subclass of LVM manifolds, we characterize those admitting an SKT metric. |
| title | Non-Kähler metrics on complex manifolds of LVMB type |
| topic | Differential Geometry Complex Variables 32Q99, 53C55, 14M25, 32L05, 32M25 |
| url | https://arxiv.org/abs/2605.28621 |