Algebraic dimension of twistor spaces and scalar curvature of anti-self- dual metrics
Let \(t:Z\to M\) be the twistor fibration of an anti-self-dual 4-manifold \((M,g)\). We prove that the algebraic dimension \(a(Z)\) of \(Z\) is at most 1 when the conformal class \([g]\) contains a metric of zero scalar curvature and that equality holds exactly when this metric is Ricci-flat. We also show that if \(a(Z)\) is at least 2 then \([g]\) contains a metric of positive scalar curvature. These are generalizations of theorems of \textit{Y. S. Poon} [Proc. Am. Math. Soc. 111, No. 2, 331-338 (1991; Zbl 0682.53067) and Math. Ann. 282, No. 4, 621-627 (1988; Zbl 0665.32014)] and together with a result of Gauduchon-Ville [\textit{P. Gauduchon}, `Les structures holomorphes du fibré tangent vertical sur l'espace des twisteur d'une variété conforme autoduale'. Preprint (1989)] give a fairly complete account on the relation between \(a(Z)\) and the sign of the scalar curvature of the conformal class \([g]\).
scientific article; zbMATH DE number 59952
scientific article