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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.10611 |
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| _version_ | 1866911551993151488 |
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| author | Wang, Mingwei Yang, Xiaokui Yau, Shing-Tung |
| author_facet | Wang, Mingwei Yang, Xiaokui Yau, Shing-Tung |
| contents | In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $ E $ be a holomorphic vector bundle over a compact Kähler manifold $(M,ω_g) $. Suppose that there exists a smooth Hermitian metric $ h_0 $ on $E$ such that the Hermitian-Yang-Mills tensor $ Λ_{ω_g}\sqrt{-1} R^{h_0} $ is positive definite. Then for any Hermitian positive definite tensor $ P\in Γ\left(M,E^*\otimes \overline E^*\right) $, there exists a unique smooth Hermitian metric $ h $ on $E$ such that $$Λ_{ω_g} \sqrt{-1} R^h=P.$$ The proof is based on a new comparison theorem for Hermitian-Yang-Mills tensors. Inspired by these results, we have also derived quantitative Chern number inequalities that apply to both holomorphic vector bundles and compact Kähler manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10611 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | RC-positivity, comparison theorems and prescribed Hermitian-Yang-Mills tensors I Wang, Mingwei Yang, Xiaokui Yau, Shing-Tung Differential Geometry 53C55 In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $ E $ be a holomorphic vector bundle over a compact Kähler manifold $(M,ω_g) $. Suppose that there exists a smooth Hermitian metric $ h_0 $ on $E$ such that the Hermitian-Yang-Mills tensor $ Λ_{ω_g}\sqrt{-1} R^{h_0} $ is positive definite. Then for any Hermitian positive definite tensor $ P\in Γ\left(M,E^*\otimes \overline E^*\right) $, there exists a unique smooth Hermitian metric $ h $ on $E$ such that $$Λ_{ω_g} \sqrt{-1} R^h=P.$$ The proof is based on a new comparison theorem for Hermitian-Yang-Mills tensors. Inspired by these results, we have also derived quantitative Chern number inequalities that apply to both holomorphic vector bundles and compact Kähler manifolds. |
| title | RC-positivity, comparison theorems and prescribed Hermitian-Yang-Mills tensors I |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2603.10611 |