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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
##### Keywords:
Hopf surface; twistor space; algebraic reduction