Locally conformally Kähler metrics on Hopf surfaces

Gauduchon, Paul; Ornea, Liviu

Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 1107-1127.

Une surface de Hopf primaire est une surface complexe compacte dont le revêtement universel est ${ℂ}^2-\left\{\right(0,0\left)\right\}$ et dont le groupe fondamental est le groupe cyclique engendré par une transformation $(u,v)\mapsto (\alpha u+\lambda v^m,\beta v)$, $m\in \mathbb{Z}$, pour $\alpha ,\beta ,\lambda \in ℂ$ tels que $\mid \alpha \mid \ge \mid \beta \mid >1$ et $(\alpha -{\beta }^m)\lambda =0$. 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 $\lambda =0$ et $\left|\alpha \right|=\left|\beta \right|$. Nous construisons ici une métrique localement conformément kählérienne, à forme de Lee parallèle, sur toute surface de Hopf primaire de la classe $1$ ($\lambda =0$). 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 ${ℂ}^2-\left\{\right(0,0\left)\right\}$ and cyclic fundamental group generated by the transformation $(u,v)\mapsto (\alpha u+\lambda v^m,\beta v)$, $m\in \mathbb{Z}$, and $\alpha ,\beta ,\lambda \in ℂ$ such that $\mid \alpha \mid \ge \mid \beta \mid >1$ and $(\alpha -{\beta }^m)\lambda =0$. Being diffeomorphic with $S^3\times S^1$ Hopf surfaces cannot admit any Kähler metric. However, it was known that for $\lambda =0$ and $\mid \alpha \mid =\mid \beta \mid$ 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 $1$ ($\lambda =0$). 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.

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