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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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