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Classification of singular germs of mappings and deformations of compact surfaces of class VII$$_0$$. (English) Zbl 0930.32013
A minimal compact complex surface $$S$$ is called a surface of class $$\text{VII}_0$$ if $$b_1(S)=0$$. The Hopf surfaces belong to this category. Surface of class $$\text{VII}_0$$ with algebraic dimension $$a(S)=1$$ or with $$a(S) =0=b_2(S)$$ are well understood [see K. Kodaira, Am. J. Math. 88, 682-721 (1966; Zbl 0193.37701), F. A. Bogomolov, Math. USSR-Izv. 21, 31-73 (1983; Zbl 0527.14029), A.-D. Teleman, Int. J. Math. 5, 253-264 (1994; Zbl 0803.53038)], but a complete classification of surfaces of class $$\text{VII}_0$$ with $$a(S)=0$$ and $$b_2(S)>0$$ is still lacking. Every known surface of this type contains a so-called “global spherical shell”, a two-dimensional annular domain in $$S$$ whose complement in $$S$$ is connected. The existence of a global spherical shell in $$S$$ forces $$S$$ to contain at least one cycle of (at most $$b_2(S))$$ rational curves. It is a main question whether each surface $$S$$ of class $$\text{VII}_0$$ with $$a(S)=0$$ and $$b_2(S)>0$$ contains a global spherical shell or contains at least one curve, which then has to be elliptic or rational.
Surfaces $$S$$ of class $$\text{VII}_0$$ with a global spherical shell and $$b_2(S)=n$$ can be constructed via germs of mappings $$F=\Pi \sigma: (\mathbb{C}^2,0)\to (\mathbb{C}^2,0)$$ where $$\sigma$$ is the germ of a biholomorphic map and $$\Pi$$ factors into a product of $$n$$ blowing ups where the base points have to be chosen suitably. The authors classify those germs where the resulting minimal surface $$S$$ contains a cycle $$\Gamma$$ of rational curves with $$\Gamma^2=0$$. They prove in particular that these surfaces admit a unique foliation which is an affine bundle outside the set of rational curves with fibers isomorphic to $$\mathbb{C}$$. They also describe the semi-universal deformations of these surfaces and calculate the group of biholomorphic automorphisms which leave the irreducible curves invariant.

##### MSC:
 32J15 Compact complex surfaces 32G05 Deformations of complex structures 32G13 Complex-analytic moduli problems 32M99 Complex spaces with a group of automorphisms
##### Keywords:
surfaces of class VII$$_0$$; germs of mappings
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