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On surfaces of class \(VII_ 0\) with global spherical shells. (English) Zbl 0536.14022

This is an announcement of the author’s study on surface of type \(VII_ 0\). The complete proof will appear in his other paper: ”Rational degeneration of \(VII_ 0\) surfaces” (preprint). A minimal compact complex surface is called of class \(VII_ 0\) if its first Betti number is equal to 1. A subset \(\Sigma\) of a compact complex surface S is called a global spherical shell (abbr. to a GSS) if \((1)\quad \Sigma \cong \{(z_ 1,z_ 2)\in {\mathbb{C}}^ 2;\quad r<| z_ 1|^ 2+| z_ 2|^ 2<R\}\) and (2) the complement S-\(\Sigma\) is connected. The latter notion is due to Ma. Kato, whose fundamental result asserts that any surface with a GSS is a deformation of an (eventually non-minimal) Hopf surface. The main result of the present article gives a new characterization of such surfaces, saying that S is a surface with a GSS if and only if it is a flat deformation of a singular surface with an irreducible rational curve as a double curve whose inverse image in the normalization \(C_ 1\coprod C_ 2\) has the properties: \(C_ 1^ 2=1\), \(C^ 2_ 2=-1\). By classifying all such possible singular surfaces and by studying curves in them, he next classifies all divisors in a surface with a GSS, which together with a result by Enoki gives a very nice characterization of known surfaces of type \(VII_ 0\) with a GSS and with \(b_ 2>0\) according to the existence of special curves on such surfaces.
Reviewer: Y.Namikawa

MSC:

14J10 Families, moduli, classification: algebraic theory
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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References:

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