Sharp systolic inequalities for Kähler manifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911698745556992 |
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| author | Tsiamis, Raphael |
| author_facet | Tsiamis, Raphael |
| contents | We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the $2$-systole and the spherical $2$-systole of compact Kähler manifolds, and the stable $2$-systole of Riemannian metrics on a general class of $\mathrm{spin}^c$ manifolds and their products. These bounds attain equality precisely for complex projective space $\mathbb{CP}^n$ equipped with the Fubini--Study metric, and admit further refinements for Fano manifolds which distinguish the complex quadric, cubic, and quartic with their canonical Kähler--Einstein structures. We also obtain an algebraic characterization of manifolds admitting Kähler metrics with non-negative total scalar curvature, which implies Gromov's rational-essentialness conjecture for Kähler metrics. Finally, we prove uniform bounds for the stable $2$-systole of $\mathrm{spin}^c$ manifolds under a general essentialness condition, as well as for the Gromov width, volume, and higher stable systoles of Kähler manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20178 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp systolic inequalities for Kähler manifolds Tsiamis, Raphael Differential Geometry Complex Variables Metric Geometry We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the $2$-systole and the spherical $2$-systole of compact Kähler manifolds, and the stable $2$-systole of Riemannian metrics on a general class of $\mathrm{spin}^c$ manifolds and their products. These bounds attain equality precisely for complex projective space $\mathbb{CP}^n$ equipped with the Fubini--Study metric, and admit further refinements for Fano manifolds which distinguish the complex quadric, cubic, and quartic with their canonical Kähler--Einstein structures. We also obtain an algebraic characterization of manifolds admitting Kähler metrics with non-negative total scalar curvature, which implies Gromov's rational-essentialness conjecture for Kähler metrics. Finally, we prove uniform bounds for the stable $2$-systole of $\mathrm{spin}^c$ manifolds under a general essentialness condition, as well as for the Gromov width, volume, and higher stable systoles of Kähler manifolds. |
| title | Sharp systolic inequalities for Kähler manifolds |
| topic | Differential Geometry Complex Variables Metric Geometry |
| url | https://arxiv.org/abs/2605.20178 |