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A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes. (English) Zbl 0979.53082
We show that there exist on $$4\mathbb{C}\mathbb{P}^2$$, 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 $$U(1)$$, (iii) the metrics are not conformally isometric to the self-dual metrics constructed by C. Le Brun [J. Differ. Geom. 34, 223-253 (1991; Zbl 0725.53067)]. These are the first examples of self-dual metrics with non-semi free $$U(1)$$-isometries on simply connected manifolds. Our proof is based on the twistor theory: we use an equivariant orbifold version of the construction of S. Donaldson and R. Friedman [Nonlinearity 2, 197-239 (1989; Zbl 0671.53029)]. We also give a rough description of the structure of the algebraic reduction of the corresponding twistor spaces.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32L25 Twistor theory, double fibrations (complex-analytic aspects) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C28 Twistor methods in differential geometry
##### Citations:
Zbl 0725.53067; Zbl 0671.53029
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