## Journal of the Mathematical Society of Japan

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

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

Nobuhiro HONDA and Mitsuhiro ITOH

#### Abstract

We show that there exist on $4C{P}^{2}$, 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\left(1\right),$$\left(iii\right)$ 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 $U\left(1\right)\text{-}$ 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/05210139

**Mathematical Reviews number (MathSciNet)**

MR1727196

**Zentralblatt MATH identifier**

0979.53082

**Subjects**

Primary: 32L25: Twistor theory, double fibrations [See also 53C28]

Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

**Keywords**

Self-dual metrics twistor spaces algebraic reduction elliptic fibration

#### 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. https://projecteuclid.org/euclid.jmsj/1213107660