The 2-systole on compact K\"ahler surfaces with positive scalar curvature

We study the 2-systole on compact K\"ahler surfaces of positive scalar curvature. For any such surface $(X,\omega)$, we prove the sharp estimate $\min_X S(\omega)\cdot\operatorname{sys}_2(\omega)\le 12\pi$, with equality if and only if $X=\mathbb{P}^2$ and $\omega$ is the Fubini-Study metric. Using the classification of positive scalar curvature K\"ahler 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 K\"ahler setting.