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Paul Gauduchon; Liviu Ornea
Locally conformally Kähler metrics on Hopf surfaces
Annales de l'institut Fourier, 48 no. 4 (1998), p. 1107-1127, doi: 10.5802/aif.1651
Article PDF | Reviews MR 2000g:53088 | Zbl 0917.53025 | 2 citations in Cedram

Résumé - Abstract

A primary Hopf surface is a compact complex surface with universal cover ${\Bbb C}^2-\{(0,0)\}$ and cyclic fundamental group generated by the transformation $(u,v)\mapsto (\alpha u + \lambda v^m, \beta v)$, $m\in {\Bbb Z}$, and $\alpha ,~ \beta ,~ \lambda ~\in {\Bbb C}$ 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 ${\Bbb C}^2$. We finally infer the existence of a locally conformally Kähler metric for all primary Hopf surfaces by a deformation argument due to C. LeBrun.

Bibliography

[1] V. APOSTOLOV, P. GAUDUCHON, The Riemannian Goldberg-Sachs Theorem, Int. J. Math., 8 (1997), 421-439.  MR 98g:53080 |  Zbl 0891.53054
[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 0718.14023
[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 57 #7444 |  Zbl 0319.53026
[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 90b:53041 |  Zbl 0676.53073
[6] L.A. CORDERO, M. FERNANDEZ, M. De LEON, Compact locally conformal Kähler nilmanifolds, Geometriae Dedicata, 21 (1986), 187-192.  MR 87j:53097 |  Zbl 0601.53035
[7] S. DRAGOMIR, L. ORNEA, Locally conformal Kähler geometry, Progress in Math., 155, Birkhäuser (1998).  MR 99a:53081 |  Zbl 0887.53001
[8] P. GAUDUCHON, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518.  MR 87a:53101 |  Zbl 0536.53066
[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. Article |  MR 97d:53048 |  Zbl 0858.53039
[10] Géométrie des surfaces K3: modules et périodes. Séminaire Palaiseau, octobre 1981-janvier 1982, Astérisque, 126 (1985).  Zbl 0547.00019
[11] R. HARVEY, H. BLAINE LAWSON, Jr, An intrinsic characterisation of Kähler manifolds, Inv. Math., 74 (1983), 139-150.  Zbl 0553.32008
[12] S. KOBAYASHI, K. NOMIZU, Foundations of differential geometry, Interscience Publishers, New York, vol. I, 1963.  MR 27 #2945 |  Zbl 0119.37502
[13] K. KODAIRA, On the structure of compact complex analytic surfaces, II, American J. Math., 88 (1966), 682-722.  MR 34 #5112 |  Zbl 0193.37701
[14] K. KODAIRA, Complex structures on S1 ȕ S3, Proc. Nat. Acad. Sci. USA, 55 (1966), 240-243.  MR 33 #4955 |  Zbl 0141.27402
[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 22 #5991 |  Zbl 0128.16902
[16] B. KOSTANT, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Amer. Math. Soc., 80 (1955), 528-542.  MR 18,930a |  Zbl 0066.16001
[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 5,15h |  Zbl 0060.38302
[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 84j:32036 |  Zbl 0557.32004
[21] S. TANNO, The standard CR structure on the unit tangent bundle, Tohoku Math. J., 44 (1992), 535-543. Article |  MR 93k:53033 |  Zbl 0779.53024
[22] F. TRICERRI, Some examples of locally conformal Kähler manifolds, Rend. Sem. Mat. Univ. Politecn. Torino, 40 (1982), 81-92.  MR 84j:53073 |  Zbl 0511.53068
[23] I. VAISMAN, Some curvature properties of Locally Conformal Kähler Manifolds, Trans. Amer. Math. Soc., 259 (1980), 439-447.  MR 81d:53044 |  Zbl 0435.53044
[24] I. VAISMAN, On locally and Globally Conformal Kähler Manifolds, Trans. Amer. Math. Soc., 262 (1980), 533-542.  MR 81j:53064 |  Zbl 0446.53048
[25] I. VAISMAN, Generalized Hopf manifolds, Geometriae Dedicata, 13 (1982), 231-255.  MR 84g:53096 |  Zbl 0506.53032
[26] I. VAISMAN, Non-Kähler metrics on geometric complex surfaces, Rend. Sem. Mat. Univ. Politecn. Torino, Vol. 45, 3 (1987), 117-123.  MR 91a:32039 |  Zbl 0696.53039
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