The moduli space of hyperbolically imbedded manifolds

In this Note, we construct the moduli space of hyperbolically imbedded manifolds. We recall that the moduli space of compact hyperbolic manifolds has been constructed by Brody and Wright. To construct our moduli space, we use 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 manifold and D is a normal crossing divisor in X such that X\D is hyperbolically imbedded in X. This criterion is based on two ingredients: in our case, the first is the existence of semi-universal logarithmic deformation due to Kawamata. The second is a consequence of a theorem of stability of hyperbolically imbedded spaces through logarithmic deformations. We use the relative-distance of Kobayashi to simplify the proof.