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Projectively flat surfaces and Bogomolov’s theorem on class \(VII_ 0\) surfaces. (English) Zbl 0803.53038

Summary: We give a complete proof of Bogomolov’s theorem on class \(VII_ 0\) surfaces starting with the idea of Li, Yau and Zheng to use Kobayashi- Hitchin correspondence. We show that, because of the non-topological character of Gauduchon’s degree, the proof of these authors is not complete. (After the submission of the paper, we received a preprint of Li, Yau and Zheng, containing the line of a complete proof.) We prove that any projectively flat Hermitian surfaces is locally conformally flat-Kähler, which reduces the problem to the classification of locally conformally flat-Kähler surfaces.

MSC:

53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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