Quasi-positive curvature and vanishing theorems
Abstract
In this paper, we consider 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, it is projective.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.03895
- arXiv:
- arXiv:2405.03895
- Bibcode:
- 2024arXiv240503895T
- Keywords:
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- Mathematics - Differential Geometry;
- 53C55
- E-Print:
- 10 pages