On Deformations of Hyperbolic Varities

TL;DR

This paper investigates the properties of flat deformations of hyperbolic real subschemes in projective space, establishing their topological connectedness and providing conditions for deforming to smooth hyperbolic varieties.

Contribution

It proves the closedness and connectedness of the hyperbolic subscheme subset in the Hilbert scheme and offers criteria for deforming hyperbolic subschemes to smooth ones.

Findings

The subset of hyperbolic subschemes is closed and connected.

Smooth hyperbolic varieties lie in the interior of this subset.

Sufficient conditions are provided for deforming hyperbolic subschemes to smooth hyperbolic varieties.

Abstract

In this paper we study flat deformations of real subschemes of $\mathbb{P}^n$, hyperbolic with respect to a fixed linear subspace, i.e. admitting a finite surjective and real fibered linear projection. We show that the subset of the corresponding Hilbert scheme consisting of such subschemes 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.