The 2-systole on compact Kähler surfaces with positive scalar curvature

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.