Classification of singular germs of mappings and deformations of compact surfaces of class VII\(_0\).

*(English)*Zbl 0930.32013A 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.

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.

Reviewer: E.Oeljeklaus (Bremen)