Sharp systolic inequalities for Kähler manifolds

We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the $2$-systole and the spherical $2$-systole of compact Kähler manifolds, and the stable $2$-systole of Riemannian metrics on a general class of ${\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ manifolds and their products. These bounds attain equality precisely for complex projective space ${\mathbb{C}\mathbb{P}}^n$ equipped with the Fubini–Study metric, and admit further refinements for Fano manifolds which distinguish the complex quadric, cubic, and quartic with their canonical Kähler–Einstein structures. We also obtain an algebraic characterization of manifolds admitting Kähler metrics with non-negative total scalar curvature, which implies Gromov’s rational-essentialness conjecture for Kähler metrics. Finally, we prove uniform bounds for the stable $2$-systole of ${\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ manifolds under a general essentialness condition, as well as for the Gromov width, volume, and higher stable systoles of Kähler manifolds.