On mixed curvature for Hermitian manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915093512454144 |
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| author | Tang, Kai |
| author_facet | Tang, Kai |
| contents | In this paper, we consider {\em mixed curvature} $\mathcal{C}_{α,β}$ for Hermitian manifolds, which is a convex combination of the first Chern Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam \cite{CLT}. We prove that if a compact Hermitian surface with constant mixed curvature $c$, then the Hermitian metric must be Kähler unless $c=0$ and $2α+β=0$, which extends a previous result by Apostolov-Davidov-Muškarov. For the higher-dimensional case, we also partially classify compact locally conformal Kähler manifolds with constant mixed curvature. Lastly, we prove that if $β\geq0, α(n+1)+2β>0$, then a compact Hermitian manifold with semi-positive but not identically zero mixed curvature has Kodaira dimension $-\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On mixed curvature for Hermitian manifolds Tang, Kai Differential Geometry In this paper, we consider {\em mixed curvature} $\mathcal{C}_{α,β}$ for Hermitian manifolds, which is a convex combination of the first Chern Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam \cite{CLT}. We prove that if a compact Hermitian surface with constant mixed curvature $c$, then the Hermitian metric must be Kähler unless $c=0$ and $2α+β=0$, which extends a previous result by Apostolov-Davidov-Muškarov. For the higher-dimensional case, we also partially classify compact locally conformal Kähler manifolds with constant mixed curvature. Lastly, we prove that if $β\geq0, α(n+1)+2β>0$, then a compact Hermitian manifold with semi-positive but not identically zero mixed curvature has Kodaira dimension $-\infty$. |
| title | On mixed curvature for Hermitian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2501.03749 |