Non-Moishezon twistor spaces of 4𝐂𝐏² with non-trivial automorphism group

Honda, Nobuhiro

doi:10.1090/S0002-9947-05-04141-3

Non-Moishezon twistor spaces of $4{\mathbf {CP}}^2$ with non-trivial automorphism group
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Trans. Amer. Math. Soc. 358 (2006), 1897-1920 Request permission

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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