SIGMA 10 (2014), 046, 25 pages      arXiv:1311.2391      https://doi.org/10.3842/SIGMA.2014.046
Contribution to the Special Issue on Progress in Twistor Theory

Scalar Flat Kähler Metrics on Affine Bundles over $\mathbb{CP}^1$

Nobuhiro Honda
Mathematical Institute, Tohoku University, Sendai, Miyagi, Japan

Received November 12, 2013, in final form April 15, 2014; Published online April 19, 2014

Abstract
We show that the total space of any affine $\mathbb{C}$-bundle over $\mathbb{CP}^1$ with negative degree admits an ALE scalar-flat Kähler metric. Here the degree of an affine bundle means the negative of the self-intersection number of the section at infinity in a natural compactification of the bundle, and so for line bundles it agrees with the usual notion of the degree.

Key words: scalar-flat Kähler metric; affine bundle; twistor space.

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References

1. Atiyah M.F., Complex fibre bundles and ruled surfaces, Proc. London Math. Soc. 5 (1955), 407-434.
2. Calderbank D.M.J., Singer M.A., Einstein metrics and complex singularities, Invent. Math. 156 (2004), 405-443, math.DG/0206229.
3. Honda N., Deformation of LeBrun's ALE metrics with negative mass, Comm. Math. Phys. 322 (2013), 127-148, arXiv:1204.4857.
4. Kishimoto T., Prokhorov Y., Zaidenberg M., Group actions on affine cones, in Affine Algebraic Geometry, CRM Proc. Lecture Notes, Vol. 54, Amer. Math. Soc., Providence, RI, 2011, 123-163, arXiv:0905.4647.
5. Kodaira K., Complex manifolds and deformation of complex structures, Classics in Mathematics, Springer-Verlag, Berlin, 2005.
6. Kronheimer P.B., The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665-683.
7. Kronheimer P.B., A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), 685-697.
8. LeBrun C., Counter-examples to the generalized positive action conjecture, Comm. Math. Phys. 118 (1988), 591-596.
9. LeBrun C., Twistors, Kähler manifolds, and bimeromorphic geometry. I, J. Amer. Math. Soc. 5 (1992), 289-316.
10. Morrow J., Kodaira K., Complex manifolds, AMS Chelsea Publishing, Providence, RI, 2006.
11. Pontecorvo M., On twistor spaces of anti-self-dual Hermitian surfaces, Trans. Amer. Math. Soc. 331 (1992), 653-661.
12. Sernesi E., Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, Vol. 334, Springer-Verlag, Berlin, 2006.
13. Suwa T., Stratification of local moduli spaces of Hirzebruch manifolds, Rice Univ. Studies 59 (1973), 129-146.
14. Viaclovsky J.A., An index theorem on anti-self-dual orbifolds, Int. Math. Res. Not. 2013 (2013), 3911-3930, arXiv:1202.0578.