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.


32J15 Compact complex surfaces
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