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Journal Articles Mathematische Annalen Year : 2012

Infinite bubbling in non-Kählerian geometry


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 $b \to b_0$. 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 $z \to 0$. 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 $\tilde E_z$ of $E_z$ in the universal cover $\tilde X_z $does converge to an effective divisor $\tilde E_0$ in $\tilde X_0$, but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of $\tilde X_0$ 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. The authors wish to thank the unnamed referee for the careful reading of the paper and for his useful suggestions.
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hal-01255757 , version 1 (14-01-2016)



Georges Dloussky, Andrei Teleman. Infinite bubbling in non-Kählerian geometry. Mathematische Annalen, 2012, 353 (4), pp.1283-1314. ⟨10.1007/s00208-011-0713-9⟩. ⟨hal-01255757⟩
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