On deformations of hyperbolic varieties

In this paper we study flat deformations of real subschemes of Pⁿ, hyperbolic with respect to a fixed linear subspace, i.e., admit-ting a finite surjective and real fibered linear projection. We show that the subset of the corresponding Hilbert scheme consisting of such sub-schemes is closed and connected in the classical topology. Every smooth variety in this set lies in the interior of this set. Furthermore, we provide sufficient conditions for a hyperbolic subscheme to admit a flat deformation to a smooth hyperbolic subscheme. This leads to new examples of smooth hyperbolic varieties.