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Infinite bubbling in non-Kählerian geometry. (English) Zbl 1252.32026
The classification of compact complex surfaces of class VII would be complete if one could prove the GSS conjecture that every minimal class VII surface $$X$$ with $$b_2(X)>0$$ contains a global spherical shell (GSS). It is equivalent to show that $$X$$ contains $$b_2(X)$$ rational curves, see [G. Dloussky, K. Oeljeklaus and M. Toma, Tohoku Math. J., II. Ser. 55, No. 2, 283–309 (2003; Zbl 1034.32012)]. Deformation theory of class VII surfaces seems to be a natural tool to attack this problem, but in holomorphic families of non-Kählerian manifolds specific phenomena may occur which are obstacles to establishing useful deformation invariants. Examples show that in a holomorphic family $$(X_b)_{b\in B}$$ of class VII surfaces the rational curves representing a fixed 2-homology class need not form a proper family. The area of a curve $$C_b\subset X_b$$ in a fixed 2-homology class may tend to infinity as $$b\rightarrow b_0$$. This is not a contradiction to the consequence of the above conjecture that all class VII surfaces in a holomorphic family have the same number of rational curves. A deeper understanding of this phenomenon seems indispensable for solving the classification problem for class VII surfaces.
The article under review makes a significant contribution to this project. It presents a detailed and thorough analysis of the area-exploding phenomenon in holomorphic families $${\mathfrak X}=(X_z)_{z\in B}$$ of class VII surfaces with $$b_2(X_z)>0$$ and GSS over the unit ball $$B$$ in $$\mathbb C^r$$. This is done by lifting homology classes from $${\mathfrak X}$$ to its universal cover $$\tilde{\mathfrak X}=(\tilde{X_z})_{z\in B}$$. The fiber $$\tilde {X_z}$$ has two ends, a pseudoconvex and a pseudoconcave end. For $$b\in B$$ let $$e_b\in H_2(X_b,\mathbb Z)$$ with $$e_b\cdot e_b=-1=\big(e_b,-c_1(X_b)\big)$$ and $$e:=(e_b)_{b\in B}$$. For any lift $$\tilde{e}$$ of $$e$$ to $$\tilde{\mathfrak X}$$ there exists an effective divisor $$\tilde{{\mathfrak E}}\subset\tilde{\mathfrak X}$$ flat over $$B$$ with the following property: If $$X_b$$ contains an exceptional curve $$E_b$$ representing $$e_b$$ then the fiber $$\tilde{E_b}\subset{\mathfrak E}$$ is a lift of $$E_b$$. If $$X_b$$ does not contain an exceptional effective divisor representing $$e_b$$ then the fiber $$\tilde{E_b}\subset{\mathfrak E}$$ is a series of compact curves escaping to infinity towards the pseudoconcave end of $$\tilde{X_b}$$. These divisors represent elements of the second Borel-Moore homology group. The authors describe the phenomenon as infinite bubbling and discuss its consequences for the classification of class VII surfaces.

##### MSC:
 32J15 Compact complex surfaces 32G05 Deformations of complex structures 32Q55 Topological aspects of complex manifolds 32Q57 Classification theorems for complex manifolds
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