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Locally conformally Kähler metrics on Kato surfaces. (English) Zbl 1227.32026
A Kato surface, or a surface with a global spherical shell, cannot be Kähler since its first Betti number is odd. In this work, the author shows that every Kato surface admits a locally conformally Kähler metric. As a consequence, the universal covering of a Kato surface is Kähler.

MSC:
32J15 Compact complex surfaces
32Q15 Kähler manifolds
32Q30 Uniformization of complex manifolds
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References:
[1] F. A. Belgun, On the metric structure of non-Kähler complex surfaces , Math. Ann. 317 (2000), 1-40. · Zbl 0988.32017
[2] M. Brunella, Locally conformally Kähler metrics on certain non-Kählerian surfaces , Math. Ann. 346 (2010), 629-639. · Zbl 1196.32015
[3] G. Dloussky, Structure des surfaces de Kato , Mém. Soc. Math. France (N.S.), no. 14 (1984). · Zbl 0543.32012
[4] M. Kato, “Compact complex manifolds containing ‘global’ spherical shells” in Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977) , Kinokuniya, Tokyo, 1978, 45-84. · Zbl 0379.32023
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