The Kähler positive scalar curvature–uniruledness conjecture

Let $(M,\omega)$ be a compact Kähler manifold of arbitrary dimension. A compact complex manifold $M$ is uniruled if every point of $M$ lies on a rational curve, meaning a smooth complex curve biholomorphic to $\mathbb{C}\mathbb{P}^1$. Kähler positive scalar curvature–uniruledness conjecture. The following conditions are equivalent: (1) $M$ has a Kähler metric for which the total scalar curvature is positive; (2) $M$ has a Kähler metric of positive scalar curvature; (3) $M$ has Kodaira dimension $-\infty$; and (4) $M$ is uniruled. This extends the relationship between positive scalar curvature of Kähler metrics and algebraic-geometric properties from complex surfaces to arbitrary dimensions; the paper discusses it as a natural higher-dimensional conjecture, while the equivalence with Riemannian positive scalar curvature is known not to persist in higher dimensions.