The moduli space of hyperbolically imbedded manifolds
In the paper under review the moduli problem of hyperbolically imbedded complex manifolds is dealt with. In the compact case, the moduli problem of Kobayashi hyperbolic manifolds was studied by \textit{R. Brody} [Trans. Am. Math. Soc. 235, 213-219 (1978; Zbl 0416.32013)] and \textit{M. Wright} [Trans. Am. Math. Soc. 232, 357-370 (1977; Zbl 0365.32016)] and hence they have constructed the moduli space of compact hyperbolic spaces. In this paper the author constructs the moduli space of hyperbolically imbedded complex manifolds. To construct this, he uses a general criterion to represent analytic functors by coarse moduli spaces due to Schumacher. The objects to deform are couples \((X,D)\), where \(X\) is a compact complex manifold and \(D\) is a divisor with simple normal crossings such that \(X\setminus D\) is hyperbolically imbedded in \(X\). This criterion is based on two ingredients: the first is the existence of semi-universal logarithmic deformation due to \textit{Y. Kawamata} [Math. Ann. 235, 247-265 (1978; Zbl 0363.32015)]. The second is a consequence of a stability of hyperbolically imbedded spaces through logarithmic deformations. In the proof, the author uses the relative pseudo-distance of Kobayashi.
hyperbolically imbedded complex manifolds
scientific article; zbMATH DE number 1854345