Geodesic
Surfaces of class $\mathrm{VII}_0$ and affine geometry
Izvestiya. Mathematics , Tome 21 (1983) no. 1, pp. 31-73.

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This paper gives a complete and detailed proof of the theorem to the effect that the only surfaces of class $\mathrm{VII}_0$ with $b_2=0$ are the Inoue–Kodaira surfaces. Besides that, it contains several results on manifolds with affine structures whose holonomy groups are commutative; in particular, the general case is reduced to the case when the holonomy group is diagonal. Bibliography: 16 titles. 
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F. A. Bogomolov. Surfaces of class $\mathrm{VII}_0$ and affine geometry. Izvestiya. Mathematics , Tome 21 (1983) no. 1, pp. 31-73. http://geodesic.mathdoc.fr/item/IM2_1983_21_1_a1/

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