Quasi-positive mixed curvature, vanishing theorems, and rational connectedness
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909884919840768 |
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| author | Tang, Kai |
| author_facet | Tang, Kai |
| contents | In this paper, we consider {\em mixed curvature} $\mathcal{C}_{a,b}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam. We prove that if a compact complex manifold $M$ admits a Kähler metric with quasi-positive mixed curvature and $3a+2b\geq0$, then it is projective. If $a,b\geq0$, then $M$ is rationally connected. As a corollary, the same result holds for $k$-Ricci curvature. We also show that any compact Kähler manifold with quasi-positive 2-scalar curvature is projective. Lastly, we generalize the result to the Hermitian case. In particular, any compact Hermitian threefold with quasi-positive real bisectional curvature have vanishing Hodge number $h^{2,0}$. Furthermore, if it is Kählerian, then it is projective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-positive mixed curvature, vanishing theorems, and rational connectedness Tang, Kai Differential Geometry 53C55 In this paper, we consider {\em mixed curvature} $\mathcal{C}_{a,b}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam. We prove that if a compact complex manifold $M$ admits a Kähler metric with quasi-positive mixed curvature and $3a+2b\geq0$, then it is projective. If $a,b\geq0$, then $M$ is rationally connected. As a corollary, the same result holds for $k$-Ricci curvature. We also show that any compact Kähler manifold with quasi-positive 2-scalar curvature is projective. Lastly, we generalize the result to the Hermitian case. In particular, any compact Hermitian threefold with quasi-positive real bisectional curvature have vanishing Hodge number $h^{2,0}$. Furthermore, if it is Kählerian, then it is projective. |
| title | Quasi-positive mixed curvature, vanishing theorems, and rational connectedness |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2405.03895 |