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

17 pages; comments welcome. v3: final version. To appear in Trans. Amer. Math. Soc

References in corpus (5)

• Effectivity in the Hyperbolicity-related problems
• Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces
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Cited by in corpus (1)

• A remark on the intersection of plane curves