Hyperbolicity of cyclic covers and complements

Abstract

We prove that a cyclic cover of a smooth complex projective variety is Brody hyperbolic if its branch divisor is a generic small deformation of a large enough multiple of a Brody hyperbolic base-point-free ample divisor. We also show the hyperbolicity of complements of those branch divisors. As an application, we find new examples of Brody hyperbolic hypersurfaces in P n + 1 \mathbb {P}^{n+1} that are cyclic covers of P n \mathbb {P}^n .