Some constructions of hyperbolic hypersurfaces in $\mathrm{P}^n (\mathrm{C})$

Abstract

We show some methods of constructing hyperbolic hypersurfaces in the complex projective space, which gives a hyperbolic hypersurface of degree $2^n$ in $P^n (\mathbf{C})$ for every $n \ge 2$. Moreover, we show that there are some hyperbolic hypersurfaces of degree $d$ in $P^n (\mathbf{C})$ for every $d \ge 2 \times 6^n$ for each $n \ge 3$.