A construction of hyperbolic hypersurface of \(P^ n(C)\) (Q1910166)

S. Kobayashi conjectured in 1970 that a generic hypersurface of large degree of the complex projective space \(\mathbb{P}^n (\mathbb{C})\) of dimension \(n\) is hyperbolic, and that its complement is hyperbolic and moreover hyperbolically embedded into \(\mathbb{P}^n (\mathbb{C})\). Even the existence of such hypersurfaces has been known only for \(n \leq 3\). In this paper we construct such hypersurfaces for all \(n \geq 2\).