A 2-systolic inequality on non-rational compact K\"ahler surfaces with positive scalar curvature

Abstract

In this note, we prove a 2-systolic inequality on compact positive scalar curvature K\"ahler surfaces admitting a nonconstant holomorphic map to a positive-genus compact Riemann surface. According to the classification of positive scalar curvature K\"ahler surfaces, any such surface must be a ruled surface fibred over a complex curve with positive genus.