The 2-systole on compact Kähler surfaces with positive scalar curvature
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914468649238528 |
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| author | Sha, Zehao |
| author_facet | Sha, Zehao |
| contents | We study the 2-systole on compact Kähler surfaces of positive scalar curvature. For any such surface $(X,ω)$, we prove the sharp estimate $\min_X S(ω)\cdot\operatorname{sys}_2(ω)\le 12π$, with equality if and only if $X=\mathbb{P}^2$ and $ω$ is the Fubini-Study metric. Using the classification of positive scalar curvature Kähler surfaces, we determine the optimal constant in each case and describe the corresponding rigid models. When $X$ is a non-rational ruled surface, we also give an independent analytic proof, adapting Stern's level set method to the holomorphic fibration in Kähler setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02901 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The 2-systole on compact Kähler surfaces with positive scalar curvature Sha, Zehao Differential Geometry We study the 2-systole on compact Kähler surfaces of positive scalar curvature. For any such surface $(X,ω)$, we prove the sharp estimate $\min_X S(ω)\cdot\operatorname{sys}_2(ω)\le 12π$, with equality if and only if $X=\mathbb{P}^2$ and $ω$ is the Fubini-Study metric. Using the classification of positive scalar curvature Kähler surfaces, we determine the optimal constant in each case and describe the corresponding rigid models. When $X$ is a non-rational ruled surface, we also give an independent analytic proof, adapting Stern's level set method to the holomorphic fibration in Kähler setting. |
| title | The 2-systole on compact Kähler surfaces with positive scalar curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2601.02901 |