Hyperbolicity of cyclic covers and complements

TL;DR

This paper establishes conditions under which cyclic covers and their complements are Brody hyperbolic, providing new examples of hyperbolic hypersurfaces in projective space through deformation techniques.

Contribution

It proves Brody hyperbolicity for cyclic covers with branch divisors near multiples of hyperbolic divisors and introduces new hyperbolic hypersurfaces in projective space.

Findings

Cyclic covers are Brody hyperbolic under certain deformations.

Complements of branch divisors are also hyperbolic.

New examples of hyperbolic hypersurfaces in projective space.

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 $\mathbb{P}^{n+1}$ that are cyclic covers of $\mathbb{P}^n$.