Locally conformally Kähler metrics on Hopf surfaces

Gauduchon, Paul; Ornea, Liviu

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

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.

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.

[1] V. Apostolov, P. Gauduchon, The Riemannian Goldberg-Sachs Theorem, Int. J. Math., 8 (1997), 421-439. | MR | Zbl

[2] W. Barth, C. Peters, A. Van De Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band, 4, Springer-Verlag, 1984. | Zbl

[3] F. Belgun, Complex surfaces admitting no metric with parallel Lee form, preprint.

[4] D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, 1976. | MR | Zbl

[5] L.C. De Andres, L.A. Cordero, M. Fernandez, J.J. Mencia, Examples of four dimensional locally conformal Kähler manifolds, Geometriae Dedicata, 29 (1989), 227-233. | MR | Zbl

[6] L.A. Cordero, M. Fernandez, M. De Leon, Compact locally conformal Kähler nilmanifolds, Geometriae Dedicata, 21 (1986), 187-192. | MR | Zbl

[7] S. Dragomir, L. Ornea, Locally conformal Kähler geometry, Progress in Math., 155, Birkhäuser (1998). | MR | Zbl

[8] P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518. | MR | Zbl

[9] P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S1 ȕ S3, J. Reine Angew. Math., 469 (1995), 1-50. | MR | Zbl

[10] Géométrie des surfaces K3: modules et périodes. Séminaire Palaiseau, octobre 1981-janvier 1982, Astérisque, 126 (1985). | Numdam | Zbl

[11] R. Harvey, H. Blaine Lawson, Jr, An intrinsic characterisation of Kähler manifolds, Inv. Math., 74 (1983), 139-150. | Zbl

[12] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Interscience Publishers, New York, vol. I, 1963. | MR | Zbl

[13] K. Kodaira, On the structure of compact complex analytic surfaces, II, American J. Math., 88 (1966), 682-722. | MR | Zbl

[14] K. Kodaira, Complex structures on S1 ȕ S3, Proc. Nat. Acad. Sci. USA, 55 (1966), 240-243. | MR | Zbl

[15] K. Kodaira, D.C. Spencer, On deformations of complex analytic structures, III, stability theorems for complex structures, Ann. of Math., 71 (1960), 43-77. | MR | Zbl

[16] B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Amer. Math. Soc., 80 (1955), 528-542. | MR | Zbl

[17] C. Lebrun, Private letter to the first named author, September 22, 1992.

[18] H.C. Lee, A kind of even dimensional differential geometry and its application to exterior calculus, American J. Math., 65 (1943), 433-438. | MR | Zbl

[19] P. Piccinni, Attempts of writing metrics on primary Hopf surfaces, private communication, October 1991.

[20] Y.-T. Siu, Every K3 surface is Kähler, Inv. Math., 73 (1983), 139-150. | MR | Zbl

[21] S. Tanno, The standard CR structure on the unit tangent bundle, Tohoku Math. J., 44 (1992), 535-543. | MR | Zbl

[22] F. Tricerri, Some examples of locally conformal Kähler manifolds, Rend. Sem. Mat. Univ. Politecn. Torino, 40 (1982), 81-92. | MR | Zbl

[23] I. Vaisman, Some curvature properties of Locally Conformal Kähler Manifolds, Trans. Amer. Math. Soc., 259 (1980), 439-447. | MR | Zbl

[24] I. Vaisman, On locally and Globally Conformal Kähler Manifolds, Trans. Amer. Math. Soc., 262 (1980), 533-542. | MR | Zbl

[25] I. Vaisman, Generalized Hopf manifolds, Geometriae Dedicata, 13 (1982), 231-255. | MR | Zbl

[26] I. Vaisman, Non-Kähler metrics on geometric complex surfaces, Rend. Sem. Mat. Univ. Politecn. Torino, Vol. 45, 3 (1987), 117-123. | MR | Zbl