Self-dual metrics and twenty-eight bitangents

Nobuhiro Honda Tokyo Institute of Technology

Differential Geometry mathscidoc:1609.10005

Journal of Differential Geometry, 75, 175-258, 2007
We determine a global structure of the moduli space of self-dual metrics on 3CP2 satisfying the following three properties: (i) the scalar curvature is of positive type, (ii) they admit a non-trivial Killing field, (iii) they are not conformal to the LeBrun’s self- dual metrics based on the ‘hyperbolic ansatz’. We prove that the moduli space of these metrics is isomorphic to an orbifold R3/G, where G is an involution of R3 having two-dimensional fixed locus. In particular, the moduli space is non-empty and connected. We also remark that Joyce’s self-dual metrics with torus symmetry appear as a limit of our self-dual metrics. Our proof of the result is based on the twistor theory. We first determine a defining equation of a projective model of the twistor space of the metric, and then prove that the projective model is always birational to a twistor space, by determining the family of twistor lines. In determining them, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.
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@inproceedings{nobuhiro2007self-dual,
  title={SELF-DUAL METRICS AND TWENTY-EIGHT BITANGENTS},
  author={Nobuhiro Honda},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908192443614860662},
  booktitle={Journal of Differential Geometry},
  volume={75},
  pages={175-258},
  year={2007},
}
Nobuhiro Honda. SELF-DUAL METRICS AND TWENTY-EIGHT BITANGENTS. 2007. Vol. 75. In Journal of Differential Geometry. pp.175-258. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908192443614860662.
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