Generalised Hermite-Einstein Fibre Metrics and Slope Stability for Holomorphic Vector Bundles
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915701453750272 |
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| author | Popovici, Dan |
| author_facet | Popovici, Dan |
| contents | Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a Kähler metric $ω$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form $Ω$. We introduce the notions of $(ω,\,Ω)$-Hermite-Einstein holomorphic vector bundles and $(ω,\,Ω)$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $ω$. We then prove that the $(ω,\,Ω)$-Hermite-Einstein condition implies the $(ω,\,Ω)$-semi-stability of a holomorphic vector bundle and its splitting into $(ω,\,Ω)$-stable subbundles. This extends a classical result by Kobayashi and Lübke to our generalised setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24932 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalised Hermite-Einstein Fibre Metrics and Slope Stability for Holomorphic Vector Bundles Popovici, Dan Algebraic Geometry Complex Variables Differential Geometry Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a Kähler metric $ω$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form $Ω$. We introduce the notions of $(ω,\,Ω)$-Hermite-Einstein holomorphic vector bundles and $(ω,\,Ω)$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $ω$. We then prove that the $(ω,\,Ω)$-Hermite-Einstein condition implies the $(ω,\,Ω)$-semi-stability of a holomorphic vector bundle and its splitting into $(ω,\,Ω)$-stable subbundles. This extends a classical result by Kobayashi and Lübke to our generalised setting. |
| title | Generalised Hermite-Einstein Fibre Metrics and Slope Stability for Holomorphic Vector Bundles |
| topic | Algebraic Geometry Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2512.24932 |