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Algebraic reduction of twistor spaces of Hopf surfaces. (English) Zbl 1002.32014
Twistor theory provides a rich supply of compact complex 3-manifolds. Even if \(S\) is a conformally flat compact 4-manifold, its associated twistor space can have a rich function theory. Consider, for example, Hopf surfaces \(S(\alpha_1,\alpha_2)\) obtained as the quotient of \({\mathbb C}^2\setminus\{0\}\) by \((z_1,z_2)\mapsto(\alpha_1 z_1,\alpha_2 z_2)\) for \(\alpha_i\in{\mathbb C}\) with \(|\alpha_1|=|\alpha_2|>1\). For generic parameters \(\alpha_i\), the corresponding twistor space \(Z=Z(\alpha_1,\alpha_2)\) has no meromorphic functions but for special values, it can have algebraic dimension \(1\) or \(2\). In case \(Z\) has algebraic dimension 1, there is a meromorphic mapping \(f:Z\to Y\) onto a nonsingular curve \(Y\). In this case, the author identifies the fibres as either Hopf surfaces or as having normalisation a nonsingular ruled surface of genus one.

MSC:
32J17 Compact complex \(3\)-folds
32L25 Twistor theory, double fibrations (complex-analytic aspects)
14J30 \(3\)-folds
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