Locally conformally Kähler metrics on Hopf surfaces

Gauduchon, Paul; Ornea, Liviu

Annales de l'Institut Fourier, Tome 48 (1998), p. 1107-1127 / Harvested from Numdam

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Abstract

Une surface de Hopf primaire est une surface complexe compacte dont le revêtement universel est $ℂ^{2} - {(0, 0)}$ et dont le groupe fondamental est le groupe cyclique engendré par une transformation $(u, v) \mapsto (α u + λ v^{m}, β v)$ , $m \in ℤ$ , pour $α, β, λ \in ℂ$ tels que $∣ α ∣ \geq ∣ β ∣ > 1$ et $(α - β^{m}) λ = 0$ . Les surfaces de Hopf primaires sont difféomorphes à $S^{3} \times S^{1}$ 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 $ℂ^{2}$ . 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.

A primary Hopf surface is a compact complex surface with universal cover $ℂ^{2} - {(0, 0)}$ and cyclic fundamental group generated by the transformation $(u, v) \mapsto (α u + λ v^{m}, β v)$ , $m \in ℤ$ , and $α, β, λ \in ℂ$ such that $∣ α ∣ \geq ∣ β ∣ > 1$ and $(α - β^{m}) λ = 0$ . Being diffeomorphic with $S^{3} \times S^{1}$ 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 $ℂ^{2}$ . We finally infer the existence of a locally conformally Kähler metric for all primary Hopf surfaces by a deformation argument due to C. LeBrun.

Publié le : 1998-01-01

MR 2000g:53088 | Zbl 0917.53025 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1651

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     author = {Gauduchon, Paul and Ornea, Liviu},
     title = {Locally conformally K\"ahler metrics on Hopf surfaces},
     journal = {Annales de l'Institut Fourier},
     volume = {48},
     year = {1998},
     pages = {1107-1127},
     doi = {10.5802/aif.1651},
     mrnumber = {2000g:53088},
     zbl = {0917.53025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1998__48_4_1107_0}
}

Gauduchon, Paul; Ornea, Liviu. Locally conformally Kähler metrics on Hopf surfaces. Annales de l'Institut Fourier, Tome 48 (1998) pp. 1107-1127. doi : 10.5802/aif.1651. http://gdmltest.u-ga.fr/item/AIF_1998__48_4_1107_0/

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