A Kummer-type construction of self-dual 4-manifolds.

*(English)*Zbl 0818.53034First, 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 |

##### References:

[1] | Anderson, M.T.: Moduli spaces of Einstein metrics on 4-manifolds. Bull. Am. Math. Soc.21, 275-279 (1989) · Zbl 0697.58057 |

[2] | Atiyah, M., Hitchin, N.J., Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. R. Soc. Lond. A362, 425-461 (1978) · Zbl 0389.53011 |

[3] | Bailey, T.N., Singer, M.A.: Twistors, massless fields, and the Penrose transform. In: Twistors in mathematics and physics (Bailey and Baston, eds.). Lond. Math. Soc. Lect. Notes 156, 1990, pp. 299-338 |

[4] | Baily, W.L.: The decomposition theorem forV-manifolds. Am. J. Math.78, 862-888 (1956) · Zbl 0173.22705 |

[5] | Donaldson, S.K., Friedman, R.D.: Connected sums of self-dual manifolds and deformations of singular spaces. Nonlinearity2, 197-239 (1989) · Zbl 0671.53029 |

[6] | Floer, A.: Self-dual conformal structures onl??2. J. Differ. Geom.33, 551-573 (1991) · Zbl 0736.53046 |

[7] | Gibbons, G.W., Pope, C.N.: The positive action conjecture and asymptotically Euclidean metrics in quantum gravity. Commun. Math. Phys.66, 267-290 (1979) |

[8] | Hitchin, N.J.: Polygons and gravitons. Math. Proc. Camb. Phil. Soc.83, 465-476 (1979) · Zbl 0405.53016 |

[9] | Kuiper, H.N.: On conformally flat spaces in the large. Ann. Math.50, 916-924 (1949) · Zbl 0041.09303 |

[10] | Kronheimer, P.B.: A Torelli-type theorem for gravitational instantons. J. Differ. Geom.29, 685-697 (1989) · Zbl 0671.53046 |

[11] | LeBrun, C.R.: Scalar-flat K?hler metrics on blown-up ruled surfaces. J. Reine Angew. Math.420, 161-177 (1991) · Zbl 0727.53067 |

[12] | LeBrun, C.R., Singer, M.A.: Existence and deformation theory for scalar-flat K?hler metrics on compact complex surfaces. Invent. Math.112, 273-313 (1993) · Zbl 0793.53067 |

[13] | Penrose, R.: Non-linear gravitons and curved twistor theory. Gen. Rel. Grav.7, 31-52 (1976) · Zbl 0354.53025 |

[14] | Ran, Z.: Deformations of maps. Lect. Notes Math.1389, 246-253 (1989) · Zbl 0708.14006 |

[15] | Satake, I.: On a generalization of the notion of manifolds. Proc. Natl. Acad. Sci. USA42, 359-363 (1956) · Zbl 0074.18103 |

[16] | Topiwala, P.: A new proof of the existence of K?hler-Einstein metrics on K3. Invent. Math.89, 425-448 (1987) · Zbl 0634.14025 |

[17] | Taubes, C.H.: The existence of anti-self-dual metrics. J. Differ. Geom.36, 163-253 (1992) · Zbl 0822.53006 |

[18] | Wall, C.T.C.: On simply connected 4-manifolds. J. Lond. Math. Soc.39, 141-149 (1964) · Zbl 0131.20701 |

[19] | Yau, S.T.: On the Ricci-curvature of a complex K?hler manifold and the complex Monge-Amp?re equations. Comment. Pure Appl. Math.31, 339-411 (1978) · Zbl 0369.53059 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.