Explicit construction of new moishezon twistor spaces

Nobuhiro Honda Tokyo Institute of Technology

Differential Geometry mathscidoc:1609.10122

Journal of Differential Geometry, 82, (2), 411-444, 2009
In this paper we explicitly construct Moishezon twistor spaces on nCP2 for arbitrary n ≥ 2 which admit a holomorphic Caction. When n = 2, they coincide with Y. Poon’s twistor spaces. When n = 3, they coincide with the ones studied by the author in [14]. When n ≥ 4, they are new twistor spaces, to the best of the author’s knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP4. It turns out that the structure of the minitwistor space is independent of n. Further, we explicitly construct a CP2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP1 ×CP1 in LeBrun’s case is replaced by our minitwistor spaces found in [15].
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  author={Nobuhiro Honda},
  booktitle={Journal of Differential Geometry},
Nobuhiro Honda. EXPLICIT CONSTRUCTION OF NEW MOISHEZON TWISTOR SPACES. 2009. Vol. 82. In Journal of Differential Geometry. pp.411-444. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911184143626946779.
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