Balancing the 2-systoles of some Kähler manifolds with positive scalar curvature

Abstract Theorems by Bray–Brendle–Neves and Zhu about positive scalar curvature metrics on products of a 2 $2$ 2 -sphere and an n -torus suggest that positive scalar curvature and appropriate topological assumptions should lead to the existence of topologically nontrivial 2-spheres of small area, which can be stated as an upper bound on the 2-systole of such manifolds. Recent progress has been made in this direction by Sha and Tsiamis under the additional Kähler assumption while Cecchini–Hirsh–Zeidler, Stryker, and Tsiamis showed similar upper bounds on the stable 2-systole using index theoretic methods. We prove here similar inequalities for some Kähler manifolds which control the relative sizes of the representatives of a well-chosen set of homology classes. For instance, on ( CP 1 × CP 1 , ω ) $(\mathbb {CP}^1\times \mathbb {CP}^1,\omega )$ left parenthesis double struck upper C upper P Superscript 1 Baseline times double struck upper C upper P Superscript 1 Baseline comma omega right parenthesis with a positive scalar curvature Kähler metric, we quantitatively show that largeness of one factor imposes smallness of the other.