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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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