On Kodaira dimension and scalar curvature in almost Hermitian geometry
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917026775171072 |
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| author | Zhou, Xianchao |
| author_facet | Zhou, Xianchao |
| contents | In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold $(M, J, g)$ in the Gray-Hervella class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy $κ(M, J)=-\infty$; or $κ(M, J)=0$, in which case $(M,J,g)$ is a Kähler Calabi-Yau manifold. The same conclusions also hold for compact Hermitian manifolds with an assumption of nonnegative mixed scalar curvature. As an important example, we study the twistor geometry of a compact anti-self-dual 4-manifold. In particular, for the twistor space with the Eells-Salamon almost complex structure, we show that the Kodaira dimension is zero. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16859 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Kodaira dimension and scalar curvature in almost Hermitian geometry Zhou, Xianchao Differential Geometry In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold $(M, J, g)$ in the Gray-Hervella class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy $κ(M, J)=-\infty$; or $κ(M, J)=0$, in which case $(M,J,g)$ is a Kähler Calabi-Yau manifold. The same conclusions also hold for compact Hermitian manifolds with an assumption of nonnegative mixed scalar curvature. As an important example, we study the twistor geometry of a compact anti-self-dual 4-manifold. In particular, for the twistor space with the Eells-Salamon almost complex structure, we show that the Kodaira dimension is zero. |
| title | On Kodaira dimension and scalar curvature in almost Hermitian geometry |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2510.16859 |