Metrics of nonnegative scalar curvature on surfaces of Kähler type (Q1378251)

It is known (see \textit{A. L. Besse} [`Einstein manifolds' (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 10, Springer Verlag, Berlin etc.) (1987; Zbl 0613.53001)]) that every compact manifold of dimension \(n\geq 3\) carries a Riemannian metric of (constant) negative scalar curvature, but in general there are obstructions for the existence of metrics of positive scalar curvature. Using Seiberg-Witten invariants, the author proves that the only compact complex surfaces \(X\) with \(\text{kod} (X)\geq 0\) and \(b_1(X)\) even, which carry a Riemannian metric of nonnegative scalar curvature, are minimal with \(\text{kod} (X)=0\). In this case, every such metric is Ricci-flat.