## Journal of the Mathematical Society of Japan

### A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes

#### Abstract

We show that there exist on $4CP^{2}$, 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 $U(1),$ $(\mathrm{iii})$ 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 $U(1)-$ isometries on simply connected manifolds. Our proof is based on the twistor theory: we use an equivariant orbifold version of the construction of Donaldson and Friedman [DF]. We also give a rough description of the structure of the algebraic reduction of the corresponding twistor spaces.

#### Article information

Source
J. Math. Soc. Japan Volume 52, Number 1 (2000), 139-160.

Dates
First available in Project Euclid: 10 June 2008

http://projecteuclid.org/euclid.jmsj/1213107660

Digital Object Identifier
doi:10.2969/jmsj/05210139

Mathematical Reviews number (MathSciNet)
MR1727196

Zentralblatt MATH identifier
0979.53082

#### Citation

HONDA, Nobuhiro; ITOH, Mitsuhiro. A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes. J. Math. Soc. Japan 52 (2000), no. 1, 139--160. doi:10.2969/jmsj/05210139. http://projecteuclid.org/euclid.jmsj/1213107660.