Infinite bubbling in non-Kählerian geometry

Abstract: In a holomorphic family $(X_b){b\in B}$ of non-K\"ahlerian 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-K\"ahler geometry is the {\it 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 $\widetilde E_z$ 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 a minimal surface with global spherical shell, it is given by an infinite series of {\it 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 {\it 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.