Balancing The 2-Systoles Of Some K{\"a}hler Manifolds With Positive Scalar Curvature

Abstract

Theorems by Bray-Brendle-Neves and Zhu about positive scalar curvature metrics on products of a 2-sphere and an n-torus suggests that positive scalar curvature suggests and appropriate topological assumptions should lead to the existence of a topologically non trivial 2-spheres of small area, which can be stated as upper bound on the 2-systole of such manifolds. Recent progress have been made in this direction by Sha and Tsiamis under the additional K{\"a}hler assumption while Checcini-Hirsh-Ziedler, Stryker and Tsiamis showed similar upper bounds on the stable 2-systole using index theoretic methods. We prove here similar inequalities for some K{\"a}hler manifold which control the relative sizes of the representatives of a well chosen set of homology classes. For instance on $(\mathbb{CP}^1x\mathbb{CP}^1 , \omega)$ with a positive scalar curvature K{\"a}hler metric we quantitavely show that largeness of one factor imposes smallness of the other one.