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On surfaces of class $$VII_ 0$$ with curves. II. (English) Zbl 0732.14019
This paper is a continuation of a series of papers of the author [e.g. part I of this paper: Invent. Math. 78, 393-443 (1984; Zbl 0575.14033)] who investigates the global structure of minimal surfaces of class $$VII_ 0$$ with positive second Betti number $$b_ 2$$. It was known that such surfaces contain at least $$b_ 2\quad rational$$ curves if they contain a global spherical shell.
One of the main goals of this paper is to investigate the converse of that result, known as Kato conjecture. In this respect, the author shows that for minimal $$VII_ 0$$ surface with $$b_ 2\quad rational$$ curves, the weighted dual graph of all its curves is the same as that of the dual graph of the maximal reduced curve on a minimal surface with a global spherical shell. Also he provides the characterization of Inoue surfaces with positive $$b_ 2$$ in terms of their Dloussky numbers.

##### MSC:
 14J29 Surfaces of general type
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##### References:
 [1] I. ENOKI, Surfaces of class VII0 with curves, Thoku Math. J. 33 (1981), 453-492. · Zbl 0476.14013 · doi:10.2748/tmj/1178229349 [2] G. DLOUSSKY, Sur les surfaces compactes de la classe VII0 contenant unecoquille spherique globale, C. R. Acad. Sci. Paris Ser. I. Math., 292 (1981), 727-730. · Zbl 0472.32019 [3] G. DLOUSSKY, Sur les courbes et champs de vecteurs globaux des surfaces analytiques de la class V0 admettant une coquille spherique globale, C. R. Acad. Sci. Paris, 295 (1982), 111-114. · Zbl 0505.32022 [4] G. DLOUSSKY, Structure des surfaces de Kato, Mem. Soc. Math. France, (N. S.) 14 (1984) · Zbl 0543.32012 · numdam:MSMF_1984_2_14__R1_0 · eudml:94835 [5] M. INOUE, New surfaces with no meromorphic functions, Proc. Internat. Congr. Math., Vancouver, 1974, Vol. 1, Canadian Math. Soc., 1975, pp. 423-426 (1974). · Zbl 0365.14010 [6] M. INOUE, Newsurfaces with nomeromorphicfunctions, II, Complex Analysis and Algebraic Geometr (Baily and Shioda eds.), Iwanami Shoten, Tokyo and Cambridge Univ. Press, Cambridge, 1977, pp. 91-106. · Zbl 0365.14011 · doi:10.1017/CBO9780511569197.007 [7] MA. KATO, Compact complex manifoldscontaining”global spherical shells”, I, Proc. Internat.Sympos Algebraic Geometry, (M. Nagata, ed.), Kyoto, 1977, Kinokuniya, Tokyo, pp. 45-84. · Zbl 0421.32010 [8] MA. KATO, Ona certainclass ofnon-algebraicnon-Kahler compact complex manifolds, Recent progres of algebraic geometry in Japan (M. Nagata, ed.), North-Holland Mathematics Studies, no. 71, Kinokuniya, Tokyo and North-Holland, Amsterdam and New York, 1977, pp. 28-50. · doi:10.1016/S0304-0208(09)70005-2 [9] Y. KAWAMATA, Ondeformations ofcompactifiable complex manifolds, Math. Ann. 235 (1978), 247-265 · Zbl 0363.32015 · doi:10.1007/BF01420124 · eudml:163152 [10] K. KODAIRA, On the structure of compact complex analytic surfaces, I-IV, Amer. J. Math. 86 (1964), 751-798; 88 (1966), 682-721; 90 (1968), 55-83; 90 (1968), 1048-1066. JSTOR: · Zbl 0137.17501 · doi:10.2307/2373157 · links.jstor.org [11] V. A. KARASNOV, Compact complex manifolds without meromorphic functions, Math. Notes. 1 (1975), 69-71. · Zbl 0321.32017 · doi:10.1007/BF01093846 [12] H. LAUFER, Versal deformation for two dimensionalpseudoconvex manifolds, Ann. Scuola Norm. Sup Pisa Cl. Sci. IV 7 (1980), 511-521. · Zbl 0512.32016 · numdam:ASNSP_1980_4_7_3_511_0 · eudml:83844 [13] I. NAKAMURA, On surfaces of class VII0 with curves, Proc. Japan Acad. Ser. A 58 (1982), 380-38 (1982); II 62 (1986), 406-409. · Zbl 0619.14023 · doi:10.3792/pjaa.62.406 [14] I. NAKAMURA, On surfaces of class V0 with curves, Invent. Math. 78 (1984), 393-443 · Zbl 0575.14033 · doi:10.1007/BF01388444 · eudml:143179 [15] I. NAKAMURA, On surfaces of class VII0 with global spherical shells. Proc. Japan Acad. Ser. A 5 (1983), 29-32. · Zbl 0536.14022 · doi:10.3792/pjaa.59.29 [16] I. Nakamura, VII0 surfaces and a duality of cusp singularities, Classification of Algebraic and Analyti Manifolds, Progr. Math. 39, Birkhauser, Boston 1983, pp. 333-378. · Zbl 0557.32005 [17] I. NAKAMURA, Towards classification of non-kahlerian complex surfaces, Sugaku Expos. Vol. 2, Amer Math. Soc., Providence, Rhode Island, 1989, 209-229. · Zbl 0685.14020 [18] T. ODA, Torus embeddings and applications (based on joint work with Katsuya Miyake), Tata Inst Fund. Res. Lectures on Math, and Phys., no. 58, Springer-Verlag, Berlin, Heidelberg, New York, 1978. · Zbl 0417.14043
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