Abstract
In a holomorphic family \({(X_b)_{b\in B}}\) of non-Kählerian compact manifolds, the holomorphic curves representing a fixed 2-homology class do not form a proper family in general. The deep source of this fundamental difficulty in non-Kähler geometry is the explosion of the area phenomenon: the area of a curve \({C_b\subset X_b}\) in a fixed 2-homology class can diverge as b → b 0. This phenomenon occurs frequently in the deformation theory of class VII surfaces. For instance it is well known that any minimal GSS surface X 0 is a degeneration of a 1-parameter family of simply blown up primary Hopf surfaces \({(X_z)_{z\in D{\setminus}\{0\}}}\) , so one obtains non-proper families of exceptional divisors \({E_z\subset X_z}\) whose area diverge as z → 0. Our main goal is to study in detail this non-properness phenomenon in the case of class VII surfaces. We will prove that, under certain technical assumptions, a lift \({\widetilde E_z}\) of E z in the universal cover \({\widetilde X_z}\) does converge to an effective divisor \({\widetilde E_0}\) in \({\widetilde X_0}\) , but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of \({\widetilde X_0}\) and that, when X 0 is a minimal surface with global spherical shell, it is given by an infinite series of compact rational curves, whose coefficients can be computed explicitly. This phenomenon—degeneration of a family of compact curves to an infinite union of compact curves—should be called infinite bubbling. We believe that such a decomposition result holds for any family of class VII surfaces whose generic fiber is a blown up primary Hopf surface. This statement would have important consequences for the classification of class VII surfaces.
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The authors wish to thank the unnamed referee for the careful reading of the paper and for his useful suggestions.
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Dloussky, G., Teleman, A. Infinite bubbling in non-Kählerian geometry. Math. Ann. 353, 1283–1314 (2012). https://doi.org/10.1007/s00208-011-0713-9
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DOI: https://doi.org/10.1007/s00208-011-0713-9