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Surfaces of the class VII\(_0\) admitting a vector field. II. (Surfaces de la classe \(\text{VII}_0\) admettant un champ de vecteurs. II.) (French) Zbl 1011.32014
Translation of the French summary: We finish the classification of compact holomorphic surfaces for which a nontrivial global holomorphic vector field exists. We prove, under this hypothesis, that every surface \({\mathcal S}\) of the class \(\text{VII}_0\) with \(b_2({\mathcal S})>0\) contains a global spherical shell. That is exactly the case where this classification was incomplete.
This article is the continuation of Part I [ibid. 75, No. 2, 255-270 (2000; Zbl 0984.32009)].

32J15 Compact complex surfaces
32Q57 Classification theorems for complex manifolds
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
Full Text: DOI
[1] Camacho et, C; Sad, P, Invariant varieties through singularities of holomorphic vector fields, Annals of Math., 115, 579-595, (1982) · Zbl 0503.32007
[2] Dloussky et, G; Oeljeklaus, K, Vector fields and foliations on surfaces of class VII_{0}, Ann. Inst. Fourier, 49, 1503-1545, (1999) · Zbl 0978.32021
[3] G. Dloussky et K. Oeljeklaus, M. Toma, Surfaces de la classe VII_{0} admettant un champ de vecteurs. Comm. Math. Helv. 75 (2000), 255-270. · Zbl 0984.32009
[4] Ch. Favre, Dynamique des applications rationnelles. Thèse, Orsay, 2000.
[5] C. Gellhaus et P. Heinzner, Komplexe Flächen mit holomorphen Vektorfeldern. Abh. Math. Sem. Hamburg 60 (1990), 37-46. · Zbl 0734.32017
[6] Hausen, J, Zur klassifikation glatter kompakter C*-flächen, Math. Ann., 301, 763-769, (1995) · Zbl 0830.32012
[7] Hirzebruch, F, Hilbert modular surfaces, Ens. Math., 19, 183-281, (1973) · Zbl 0285.14007
[8] J.-F. Mattei et R. Moussu, Holonomie et intégrales premières. Ann. Scient. ’ Ec. Norm. Sup. 4. série, t. 13 (1980), 499-523.
[9] Nakamura, I, On surfaces of class VII_{0} with curves, Invent. Math., 78, 393-443, (1984) · Zbl 0575.14033
[10] Nakamura, I, On surfaces of class VII_{0} with curves II, Tohôku Math. Jour., 42, 475-516, (1990) · Zbl 0732.14019
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