Infinite bubbling in non-Kählerian geometry

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.