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**A Kummer-type construction of self-dual 4-manifolds.**
*(English)*
Zbl 0818.53034

First, the authors introduce the notion called Taubes invariant for smooth oriented compact 4-manifolds. It is defined to be the least integer \(n\) such that, for a given 4-manifold \(M\), the connected sum \(M\# n\mathbb{C}\mathbb{P}_ 2\) admit a self-dual metric.

The existence of such a finite integer was proved by C. H. Taubes [J. Differ. Geom. 36, No. 1, 163-253 (1992)]; however, as the authors claim, its value is hardly calculable. In this paper, they prove that the Taubes invariant for \(M= \overline \mathbb{C}\mathbb{P}_ 2\) is between 2 and 14.

Second, and as the main contribution of this paper, the authors show how to construct the generalized connected sum of two self-dual orbifolds. Their argument extends really to the case of reflection orbifolds with a self-dual metric, these are manifolds with an action of \(\mathbb{Z}_ 2\) equipped with a global self-dual Riemannian metric. A notable example is the compactification of the Eguchi-Hanson manifold.

Following and modifying the argument in [S. Donaldson and R. Friedman, Nonlinearity 2, No. 2, 197-239 (1989; Zbl 0671.53029)] the authors show the existence of self-dual metrics on such a connected sum of reflection orbifolds satisfying certain cohomological conditions. Then, this is applied to a pair of a reflection orbifold and the compactified Eguchi-Hanson orbifold, thus concluding the estimate of the Taubes invariant of \(\overline \mathbb{C}\mathbb{P}_ 2\). As another application, the authors give an explicit construction of the Ricci-flat Kähler metric on a \(K3\) surface belonging to a certain class, which is to be compared with the general existence theorem by Calabi-Yau of such metrics on \(K3\)-surfaces.

The existence of such a finite integer was proved by C. H. Taubes [J. Differ. Geom. 36, No. 1, 163-253 (1992)]; however, as the authors claim, its value is hardly calculable. In this paper, they prove that the Taubes invariant for \(M= \overline \mathbb{C}\mathbb{P}_ 2\) is between 2 and 14.

Second, and as the main contribution of this paper, the authors show how to construct the generalized connected sum of two self-dual orbifolds. Their argument extends really to the case of reflection orbifolds with a self-dual metric, these are manifolds with an action of \(\mathbb{Z}_ 2\) equipped with a global self-dual Riemannian metric. A notable example is the compactification of the Eguchi-Hanson manifold.

Following and modifying the argument in [S. Donaldson and R. Friedman, Nonlinearity 2, No. 2, 197-239 (1989; Zbl 0671.53029)] the authors show the existence of self-dual metrics on such a connected sum of reflection orbifolds satisfying certain cohomological conditions. Then, this is applied to a pair of a reflection orbifold and the compactified Eguchi-Hanson orbifold, thus concluding the estimate of the Taubes invariant of \(\overline \mathbb{C}\mathbb{P}_ 2\). As another application, the authors give an explicit construction of the Ricci-flat Kähler metric on a \(K3\) surface belonging to a certain class, which is to be compared with the general existence theorem by Calabi-Yau of such metrics on \(K3\)-surfaces.

Reviewer: T.Sasaki (Kobe)

### MSC:

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

32L25 | Twistor theory, double fibrations (complex-analytic aspects) |

32S30 | Deformations of complex singularities; vanishing cycles |

### Citations:

Zbl 0671.53029
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\textit{C. LeBrun} and \textit{M. Singer}, Math. Ann. 300, No. 1, 165--180 (1994; Zbl 0818.53034)

### References:

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