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- Title
Infinite bubbling in non-Kählerian geometry.
- Authors
Dloussky, Georges; Teleman, Andrei
- 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. This phenomenon occurs frequently in the deformation theory of class VII surfaces. For instance it is well known that any minimal GSS surface X 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 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 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.
- Subjects
MANIFOLDS (Mathematics); HOLOMORPHIC functions; MAXIMA &; minima; DIFFERENTIAL geometry; H-spaces
- Publication
Mathematische Annalen, 2012, Vol 353, Issue 4, p1283
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-011-0713-9