### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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