# zbMATH — the first resource for mathematics

Structures affines et projectives sur les surfaces complexes. (Affine and projective structures on complex surfaces). (French) Zbl 0920.32027
It is known by works of M. Inoue, S. Kobayashi and T. Ochiai [J. Fac. Sci. Univ. Tokyo, Sect. I A 27, 247-264 (1980; Zbl 0467.32014)] that admitting a holomorphic connection is biholomorphically to either a complex torus or a Kodaira’s surface or a Hopf’s affine surface or an Inoue’s surface or a holomorphic principal fiber over a closed Riemann surface of genus greater or equal to two, with odd first Betti number. They prove that such a surface has a complex affine structure. The author proceeds to describe in a geometric way all complex affine surfaces and for each fixed one all such structures which are compatible with its analytic structure.

##### MSC:
 32J15 Compact complex surfaces 57M50 General geometric structures on low-dimensional manifolds
Full Text:
##### References:
 [1] W. BARTH, C. PETERS, A. VAN DE VEN, Compact complex surfaces, Ergebnisse der Mathematik, vol. 4, 1984, Springer Verlag. · Zbl 0718.14023 [2] Y. BENOIST, Nilvariétés projectives, Comment. Math. Helvetici, 69 (1994), 447-473. · Zbl 0839.53033 [3] A. BLANCHARD, Automorphismes des variétés fibrées analytiques complexes, C.R.A.S., Paris, 233 (1951), 1337-1339. · Zbl 0054.07301 [4] F.A. BOGOMOLOV, Surfaces of class VII0 and affine geometry, Math. USSR Izvestiya, vol. 21 (1983), 31-73. · Zbl 0527.14029 [5] C. EHRESMANN, Sur LES espaces localement homogènes, Enseign. Math., 35 (1936), 317-333. · JFM 62.1473.03 [6] D. FRIED, W. GOLDMAN, M.W. HIRSCH, Affine manifolds with nilpotent holonomy, Comment. Math. Helvetici, 56 (1981), 487-523. · Zbl 0516.57014 [7] P. GRIFFITHS, J. HARRIS, Principles of algebraic geometry, Wiley & Sons, 1978. · Zbl 0408.14001 [8] R.C. GUNNING, Lectures on Riemann surfaces, Princeton Mathematical Notes (1966). · Zbl 0175.36801 [9] R.C. GUNNING, Lectures on vector bundles over Riemann surfaces, Princeton Mathematical Notes (1967). · Zbl 0163.31903 [10] R.C. GUNNING, Special coordinate coverings of Riemann surfaces, Math. Annalen, 170 (1970), 67-86. · Zbl 0144.33501 [11] R.C. GUNNING, Affine and projective structures on Riemann surfaces, in “Riemann surfaces and related topics”, Proceedings of the 1978 Stony Brook conference, Princeton University Press (1980). · Zbl 0457.30037 [12] M. INOUE, On surfaces of class VII0, Invent. Math., 24 (1974), 269-310. · Zbl 0283.32019 [13] M. INOUE, S. KOBAYASHI, T. OCHIAI, Holomorphic affine connections on compact complex surfaces, J. Fac. Sci. Univ. Tokyo, 27 (1980), 247-264. · Zbl 0467.32014 [14] M. KAPOVICH, On monodromy of complex projective structures, Invent. Math., 119-2 (1995), 243-265. · Zbl 0839.57011 [15] S. KOBAYASHI, T. OCHIAI, Holomorphic projective structures on compact complex surfaces, Math. Annalen, 249 (1980), 75-94. · Zbl 0422.32023 [16] K. KODAIRA, On compact analytic surfaces I : Ann. Math., 71 (1960), 111-152, II : ibid, 77 (1963), 563-626, III : ibid., 78 (1963), 1-40. · Zbl 0171.19601 [17] K. KODAIRA, On the structure of compact complex analytic surfaces I : Am. J. Math., 86 (1964), 751-798, II : ibid., 88 (1966), 682-721, III : ibid., 78 (1968), 55-83, IV : ibid., 90 (1968), 1048-1066. · Zbl 0137.17501 [18] K. MAEHARA, On elliptic surfaces whose first Betti numbers are odd, Intl. Symp. on Alg. Geom., Kyoto (1977), 565-574. · Zbl 0407.14015 [19] T. SUWA, Compact quotients of ℂ2 by affine transformation groups, J. Diff. Geometry, 10 (1975), 239-252. · Zbl 0311.57025 [20] W. THURSTON, The geometry and topology of 3-manifolds, Lecture Notes from Princeton Univ. (1977-1978). [21] A. VITTER, Affine structures on compact complex manifolds, Invent. Math., 17 (1972), 231-244. · Zbl 0253.32006 [22] C.T.C. WALL, Geometric structures on compact complex analytic surfaces, Topology, vol. 25-2 (1986), 119-153. · Zbl 0602.57014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.