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

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.