We show that there exist twistor spaces over 3CP2, the connected sum of three complex projective planes, whose half-anticanonical system gives a double covering map onto CP3 branched along a quartic surface with not only ordinary double points, but other isolated singularities. We also show that if the quartic surface is in the most degenerate form, the twistor space admits a non-trivial C*-action. Correspondingly, 3CP2 admits a self-dual metric of positive scalar curvature with U(1)-symmetry, other than LeBrun's famous metrics. This leads us to a classification of twistor spaces over 3CP2 whose automorphism group is non-trivial (or equivalantly, self-dual metrics on 3CP2 whose isometric group is non-trivial).
ASJC Scopus subject areas
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty