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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.10845 |
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| _version_ | 1866918244533665792 |
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| author | Wu, Kuang-Ru |
| author_facet | Wu, Kuang-Ru |
| contents | The concept of RC-positivity and uniform RC-positivity is introduced by Xiaokui Yang to solve a conjecture of Yau on projectivity and rational connectedness of a compact Kähler manifold with positive holomorphic sectional curvature. Some main theorems in Yang's proof hold under a weaker condition called weak RC-positivity. It is therefore natural to ask if (uniform) weak RC-positivity implies (uniform) RC-positivity. Another motivation for studying this problem is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$.
In this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform RC-positivity of direct image bundles Wu, Kuang-Ru Differential Geometry Complex Variables The concept of RC-positivity and uniform RC-positivity is introduced by Xiaokui Yang to solve a conjecture of Yau on projectivity and rational connectedness of a compact Kähler manifold with positive holomorphic sectional curvature. Some main theorems in Yang's proof hold under a weaker condition called weak RC-positivity. It is therefore natural to ask if (uniform) weak RC-positivity implies (uniform) RC-positivity. Another motivation for studying this problem is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. In this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$. |
| title | Uniform RC-positivity of direct image bundles |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2512.10845 |