Degenerate double solids as twistor spaces (Q1811411)

A metric on a four-manifold is called self-dual if the anti-self-dual part of the Weyl tensor vanishes. In this case the corresponding twistor space is a complex manifold. A closed four-manifold with a self-dual metric of positive scalar curvature is homeomorphic to a sphere or to a connected sum of \(n\) copies of \({\mathbb C} P ^ n\). By results of \textit{Y. S. Poon} [J. Differ. Geom. 24, 97--132 (1986; Zbl 0583.53054), ibid. 36, No. 2, 451--491 (1992; Zbl 0742.53024)], for a sphere or \({\mathbb C}P ^ 1\) such a metric has to be a standard one and for \(n=2\) to such a metric belongs an explicit one-dimensional family. In the present paper the author considers the next case \(n=3\). It is known that such a twistor space is either a LeBrun space or a small resolution of a quartic double solid. For the last case \textit{B. Kreussler} and \textit{H. Kurke} [Compos. Math. 82, No. 1, 25--55 (1992; Zbl 0766.53049)] proved the existence of a twistor space in a generic case and gave the list of possible degenerations of a branching quartic surface. In the present paper the existence of twistor spaces is established for all degenerations from this list. The author also proves that a twistor space admits an effective \(\mathbb C ^ \ast\) action if and only if it is a twistor space of the second kind with the most degenerate branching surface or a LeBrun space.