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Class VII$$_0$$ surfaces with $$b_2$$ curves. (English) Zbl 1034.32012
This paper is devoted to the proof of Kato’s conjecture: If $$S$$ is a class VII$$_0$$ surface with the second Betti number $$b_2(S) > 0$$ rational curves (special surface by Nakamura), then $$S$$ admits global spherical shells. The authors prove that if $$S$$ is special, then the canonical bundle of a suitable finite ramified covering of $$S$$ is numerically divisorial. Using the knowledge of the configuration of rational curves on a special surface, the existence of a logarithmic 1-form twisted by a flat line bundle is proved. Passing to a finite ramified covering, a global twisted holomorphic vector field is got. The twisting is again by some flat line bundle. This induces a true holomorphic vector field on the universal covering $$\widetilde S$$ of $$S$$. This holomorphic vector field is completely integrable and the universal covering of the complement of the curves of $$S$$ is isomorphic to $$\mathbb H\times\mathbb C$$, where $$\mathbb H$$ denotes the complex half plane. The action of the fundamental group on $$\mathbb H\times\mathbb C$$ is computed. This allows one to recover the contracting rigid germ of holomorphic map which gave birth to the surface $$S$$. Using the result of Ch. Favre, which classifies such germs, the authors conclude the proof of Kato’s conjecture.

MSC:
 32J15 Compact complex surfaces
Full Text:
References:
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