The 2-systole on compact Kähler surfaces with positive scalar curvature
Zehao Sha
TL;DR
The paper establishes sharp 2-systolic inequalities for PSC Kähler surfaces by connecting holomorphic geometry with scalar curvature via the scale-invariant functional \\mathcal{J}_X([\\omega]). It combines an algebraic mass-shift optimization on the Kähler cone (after blow-ups) with an analytic Stern-level-set method for non-rational ruled surfaces to identify optimal constants according to the minimal model: 12\\pi for P^2, 8\\pi for Hirzebruch-type rulings, and 4\\pi for non-rational ruled surfaces. Consequently, the main result is a precise bound \\min_X S(\\omega) \\cdot \\text{syst}_2(\\omega) \\le 12\\pi, with equality only for (\\mathbb{P}^2, \\omega_{FS}), and a refined breakdown by minimal model yielding 12\\pi, 8\\pi, or 4\\pi in the three cases. The approaches provide a robust framework for higher-dimensional generalizations and raise natural questions about universal bounds in PSC Kähler geometry and possible higher-dimensional analogues of holomorphic systoles. The work strengthens the bridge between complex-analytic geometry, intersection theory, and global Riemannian geometry by showing how complex-analytic calibrations and Kähler cone geometry govern scalar-curvature–systole interactions.
Abstract
We study the 2-systole on compact Kähler surfaces of positive scalar curvature. For any such surface $(X,ω)$, we prove the sharp estimate \(\min_X S(ω)\cdot\syst_2(ω)\le12π\), with equality if and only if $X=\PP^2$ and $ω$ is the Fubini--Study metric. Using the classification of positive scalar curvature Kähler surfaces by their minimal models, we also determine the optimal constant in each case and describe the corresponding rigid models: $12π$ when the minimal model is $\PP^2$, $8π$ for Hirzebruch surfaces, and $4π$ for non-rational ruled surfaces. In the non-rational ruled case, we also give an independent analytic proof, adapting Stern's level set method to the holomorphic fibration in Kähler setting.