We show that there exist on , the connected sum of four complex projective planes, self-dual metrics with the following properties: (i) the sign of the scalar curvature is positive, (ii) the identity component of the isometry group is the metrics are not conformally isometric to the self-dual metrics constructed by LeBrun [LB1]. These are the first examples of self-dual metrics with non semi-free isometries on simply connected manifolds. Our proof is based on the twistor theory: we use an equivariant orbifold version of the construction of Donaldson and Friedman [DF]. We also give a rough description of the structure of the algebraic reduction of the corresponding twistor spaces.
Nobuhiro HONDA. Mitsuhiro ITOH. "A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes." J. Math. Soc. Japan 52 (1) 139 - 160, January, 2000. https://doi.org/10.2969/jmsj/05210139