Uniruledness and the sign of total scalar curvature

Abstract

For every integer $n\ge3$, we construct a smooth projective manifold $X$ of complex dimension $n$ whose canonical bundle is not pseudoeffective, or equivalently, which is uniruled, but every K\"ahler metric has negative total scalar curvature. In particular, $X$ admits no K\"ahler metric of positive scalar curvature, while it admits a Riemannian metric of positive scalar curvature. Thus the equivalence between uniruledness and the existence of a K\"ahler metric with positive scalar curvature holds in complex dimensions one and two, but fails in higher dimensions.