Deformation of a Hyperbolic embedding into the Projective Toric Variety.

In this paper, we prove the deformation theorems of algebraic divisors on an algebraic torus whose complements are Kobayashi hyperbolically embedded into a toric projective variety. Precisely, for a projective toric variety $X$, we find a Zariski closed subset inside the linear system of the divisor such that whenever $D$ is outside the Zariski closed subset, $T_N \setminus D \to X$ is a Kobayashi hyperbolic embedding. We find a deformation preserving the hyperbolicity whose parameter lives in $\mathbb{P}^1 \setminus E$ with $E$ being a finite subset. We also investigate the certain type of deformation, which we will call "similar deformations," to compute the upper bound of the number of elements of the set $E$ of deformations between two complements of divisors.