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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2508.00820 |
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| _version_ | 1866918110486855680 |
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| author | Wu, Kuang-Ru |
| author_facet | Wu, Kuang-Ru |
| contents | Let $E\to X$ be a vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. It is known that if the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ is positive, then $E\otimes \det E$ is Nakano positive by the work of Berndtsson. In this paper, we give a subharmonic analogue. Let $p:P(E^*)\to X$ be the projection and $α$ be a Kähler form on $X$. If the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $Θ$ positive on every fiber and $Θ^r\wedge p^*α^{n-1}> 0$, then $E\otimes \det E$ carries a Hermitian metric whose mean curvature is positive.
As an application, we show that the following subharmonic analogue of the Griffiths conjecture is true: if the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $Θ$ positive on every fiber and $Θ^r\wedge p^*α^{n-1}> 0$, then $E$ carries a Hermitian metric with positive mean curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean curvature of direct image bundles Wu, Kuang-Ru Differential Geometry Complex Variables Let $E\to X$ be a vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. It is known that if the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ is positive, then $E\otimes \det E$ is Nakano positive by the work of Berndtsson. In this paper, we give a subharmonic analogue. Let $p:P(E^*)\to X$ be the projection and $α$ be a Kähler form on $X$. If the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $Θ$ positive on every fiber and $Θ^r\wedge p^*α^{n-1}> 0$, then $E\otimes \det E$ carries a Hermitian metric whose mean curvature is positive. As an application, we show that the following subharmonic analogue of the Griffiths conjecture is true: if the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $Θ$ positive on every fiber and $Θ^r\wedge p^*α^{n-1}> 0$, then $E$ carries a Hermitian metric with positive mean curvature. |
| title | Mean curvature of direct image bundles |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2508.00820 |