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 $(X,\omega )$, we prove the sharp estimate ${\min }_{X}S(\omega )\cdot {\mathrm{sys}}_2(\omega )\le 12\pi$, with equality if and only if $X={\mathbb{P}}^2$ and $\omega$ is the Fubini-Study metric. Using the classification of positive scalar curvature 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.

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