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Non-Moishezon twistor spaces of $ 4{\mathbf{CP}}^2$ with non-trivial automorphism group


Author: Nobuhiro Honda
Journal: Trans. Amer. Math. Soc. 358 (2006), 1897-1920
MSC (2000): Primary 32L25, 32G05, 32G07, 53A30, 53C25
DOI: https://doi.org/10.1090/S0002-9947-05-04141-3
Published electronically: December 20, 2005
MathSciNet review: 2197434
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Abstract: We show that a twistor space of a self-dual metric on $ 4{\mathbf{CP}}^2$ with $ U(1)$-isometry is not Moishezon iff there is a $ \mathbf{C}^*$-orbit biholomorphic to a smooth elliptic curve, where the $ \mathbf C^*$-action is the complexification of the $ U(1)$-action on the twistor space. It follows that the $ U(1)$-isometry has a two-sphere whose isotropy group is $ \mathbf Z_2$. We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.


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Additional Information

Nobuhiro Honda
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi Hiroshima, 739-8526, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, 152-8551, Japan
Email: honda@math.titech.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-05-04141-3
Keywords: Self-dual metric, connected sum, twistor space, Moishezon manifold, elliptic curve
Received by editor(s): January 22, 2003
Published electronically: December 20, 2005
Additional Notes: This work was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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