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Scalar Flat Kähler Metrics on Affine Bundles over $\mathbb{CP}^1$
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: scalar-flat Kähler metric; affine bundle; twistor space. 
@article{SIGMA_2014_10_a45,
     author = {Nobuhiro Honda},
     title = {Scalar {Flat} {K\"ahler} {Metrics} on {Affine} {Bundles} over $\mathbb{CP}^1$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a45/}
}
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%J Symmetry, integrability and geometry: methods and applications
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Nobuhiro Honda. Scalar Flat Kähler Metrics on Affine Bundles over $\mathbb{CP}^1$. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a45/

[1] Atiyah M. F., “Complex fibre bundles and ruled surfaces”, Proc. London Math. Soc., 5 (1955), 407–434 | DOI | MR | Zbl

[2] Calderbank D. M. J., Singer M. A., “Einstein metrics and complex singularities”, Invent. Math., 156 (2004), 405–443, arXiv: math.DG/0206229 | DOI | MR | Zbl

[3] Honda N., “Deformation of {L}e{B}run's {ALE} metrics with negative mass”, Comm. Math. Phys., 322 (2013), 127–148, arXiv: 1204.4857 | DOI | MR | Zbl

[4] Kishimoto T., Prokhorov Y., Zaidenberg M., “Group actions on affine cones”, Affine Algebraic Geometry, CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011, 123–163, arXiv: 0905.4647 | MR | Zbl

[5] Kodaira K., Complex manifolds and deformation of complex structures, Classics in Mathematics, Springer-Verlag, Berlin, 2005 | MR | Zbl

[6] Kronheimer P. B., “The construction of {ALE} spaces as hyper-{K}ähler quotients”, J. Differential Geom., 29 (1989), 665–683 | MR | Zbl

[7] Kronheimer P. B., “A {T}orelli-type theorem for gravitational instantons”, J. Differential Geom., 29 (1989), 685–697 | MR | Zbl

[8] LeBrun C., “Counter-examples to the generalized positive action conjecture”, Comm. Math. Phys., 118 (1988), 591–596 | DOI | MR | Zbl

[9] LeBrun C., “Twistors, {K}ähler manifolds, and bimeromorphic geometry, I”, J. Amer. Math. Soc., 5 (1992), 289–316 | DOI | MR | Zbl

[10] Morrow J., Kodaira K., Complex manifolds, AMS Chelsea Publishing, 2006 | MR | Zbl

[11] Pontecorvo M., “On twistor spaces of anti-self-dual {H}ermitian surfaces”, Trans. Amer. Math. Soc., 331 (1992), 653–661 | DOI | MR | Zbl

[12] Sernesi E., Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, 334, Springer-Verlag, Berlin, 2006 | MR | Zbl

[13] Suwa T., “Stratification of local moduli spaces of {H}irzebruch manifolds”, Rice Univ. Studies, 59 (1973), 129–146 | MR | Zbl

[14] Viaclovsky J. A., “An index theorem on anti-self-dual orbifolds”, Int. Math. Res. Not., 2013 (2013), 3911–3930, arXiv: 1202.0578 | DOI | MR