Locally conformally Kähler metrics on Hopf surfaces
Annales de l'Institut Fourier, Volume 48 (1998) no. 4, p. 1107-1127

A primary Hopf surface is a compact complex surface with universal cover 2 -{(0,0)} and cyclic fundamental group generated by the transformation (u,v)(αu+λv m ,βv), m, and α,β,λ such that αβ>1 and (α-β m )λ=0. Being diffeomorphic with S 3 ×S 1 Hopf surfaces cannot admit any Kähler metric. However, it was known that for λ=0 and α=β they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class 1 (λ=0). We also show that these metrics are obtained via a Riemannian suspension over S 1 , by deforming the canonical Sasakian structure of S 3 by a Hermitian quadratic form of 2 . We finally infer the existence of a locally conformally Kähler metric for all primary Hopf surfaces by a deformation argument due to C. LeBrun.

Une surface de Hopf primaire est une surface complexe compacte dont le revêtement universel est 2 -{(0,0)} et dont le groupe fondamental est le groupe cyclique engendré par une transformation (u,v)(αu+λv m ,βv), m, pour α,β,λ tels que αβ>1 et (α-β m )λ=0. Les surfaces de Hopf primaires sont difféomorphes à S 3 ×S 1 et n’admettent donc aucune métrique kählérienne. En revanche, il est bien connu qu’elles admettent des métriques localement conformément kählériennes, à forme de Lee parallèle, dans le cas où λ=0 et |α|=|β|. Nous construisons ici une métrique localement conformément kählérienne, à forme de Lee parallèle, sur toute surface de Hopf primaire de la classe 1 (λ=0). Nous montrons aussi que ces métriques sont obtenues, via une suspension riemannienne au-dessus de S 1 , en déformant la structure sasakienne canonique de S 3 par une forme quadratique hermitienne de 2 . Finalement, nous déduisons l’existence de métriques localement conformément kählériennes sur toute surface de Hopf primaire à l’aide d’un argument de déformation dû à C. LeBrun.

@article{AIF_1998__48_4_1107_0,
     author = {Gauduchon, Paul and Ornea, Liviu},
     title = {Locally conformally K\"ahler metrics on Hopf surfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {4},
     year = {1998},
     pages = {1107-1127},
     doi = {10.5802/aif.1651},
     zbl = {0917.53025},
     mrnumber = {2000g:53088},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1998__48_4_1107_0}
}
Gauduchon, Paul; Ornea, Liviu. Locally conformally Kähler metrics on Hopf surfaces. Annales de l'Institut Fourier, Volume 48 (1998) no. 4, pp. 1107-1127. doi : 10.5802/aif.1651. http://www.numdam.org/item/AIF_1998__48_4_1107_0/

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