Structures affines et projectives sur les surfaces complexes
Annales de l'Institut Fourier, Volume 48 (1998) no. 2, p. 441-477
A complex affine (resp. projective) structure on a complex surface is an atlas of charts with value in 2 (resp. in P 2 ) with change of coordinates in the affine group A(2,) (resp. PGL(3,)). In this paper we classify the affine complex surfaces and calculate, given a complex surface S, the space of deformation of affine structures on S compatible with its analytic structure. We also show that a projective structure on a complex surface admitting an affine structure is necessarily affine.
Une structure complexe affine (resp. projective) sur une surface complexe est la donnée d’un atlas de cartes à valeur dans 2 (resp. P 2 ) à changements de cartes localement constants dans le groupe affine A(2,) (resp. le groupe PGL(3,)). Dans cet article nous classifions les surfaces complexes affines et calculons, à surface complexe S fixée, l’espace de déformation des structures complexes affines sur S compatibles avec sa structure analytique. Nous montrons aussi que toute structure projective sur une surface complexe admettant une structure complexe affine est nécessairement affine.
@article{AIF_1998__48_2_441_0,
     author = {Klingler, Bruno},
     title = {Structures affines et projectives sur les surfaces complexes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {2},
     year = {1998},
     pages = {441-477},
     doi = {10.5802/aif.1625},
     mrnumber = {99c:32038},
     zbl = {0920.32027},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_1998__48_2_441_0}
}
Structures affines et projectives sur les surfaces complexes. Annales de l'Institut Fourier, Volume 48 (1998) no. 2, pp. 441-477. doi : 10.5802/aif.1625. https://aif.centre-mersenne.org/item/AIF_1998__48_2_441_0/

[1] W. Barth, C. Peters, A. Van De Ven, Compact complex surfaces, Ergebnisse der Mathematik, vol. 4, 1984, Springer Verlag. | MR 86c:32026 | Zbl 0718.14023

[2] Y. Benoist, Nilvariétés projectives, Comment. Math. Helvetici, 69 (1994), 447-473. | MR 95m:57054 | Zbl 0839.53033

[3] A. Blanchard, Automorphismes des variétés fibrées analytiques complexes, C.R.A.S., Paris, 233 (1951), 1337-1339. | MR 13,674a | 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 | Zbl 0015.39404

[6] D. Fried, W. Goldman, M.W. Hirsch, Affine manifolds with nilpotent holonomy, Comment. Math. Helvetici, 56 (1981), 487-523. | MR 83h:53062 | Zbl 0516.57014

[7] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley & Sons, 1978. | MR 80b:14001 | Zbl 0408.14001

[8] R.C. Gunning, Lectures on Riemann Surfaces, Princeton Mathematical Notes (1966). | MR 34 #7789 | Zbl 0175.36801

[9] R.C. Gunning, Lectures on vector bundles over Riemann surfaces, Princeton Mathematical Notes (1967). | MR 37 #5888 | Zbl 0163.31903

[10] R.C. Gunning, Special coordinate coverings of Riemann surfaces, Math. Annalen, 170 (1970), 67-86. | MR 34 #7790 | 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. | MR 49 #7479 | 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. | MR 82g:32033 | Zbl 0467.32014

[14] M. Kapovich, On monodromy of complex projective structures, Invent. Math., 119-2 (1995), 243-265. | MR 95k:57017 | Zbl 0839.57011

[15] S. Kobayashi, T. Ochiai, Holomorphic projective structures on compact complex surfaces, Math. Annalen, 249 (1980), 75-94. | MR 81g:32021 | 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. | MR 32 #4708 | Zbl 0137.17501

[18] K. Maehara, On elliptic surfaces whose first Betti numbers are odd, Intl. Symp. on Alg. Geom., Kyoto (1977), 565-574. | MR 82a:32036 | Zbl 0407.14015

[19] T. Suwa, Compact quotients of ℂ2 by affine transformation groups, J. Diff. Geometry, 10 (1975), 239-252. | MR 51 #5977 | 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. | MR 49 #3761 | Zbl 0253.32006

[22] C.T.C. Wall, Geometric structures on compact complex analytic surfaces, Topology, vol. 25-2 (1986), 119-153. | MR 88d:32038 | Zbl 0602.57014