Oka properties of some hypersurface complements

TL;DR

This paper investigates the Oka properties of complements of hypersurfaces in projective and affine spaces, providing classifications for hyperplane arrangements and results for certain meromorphic function graphs.

Contribution

It characterizes which hyperplane arrangement complements are Oka and explores Oka properties of hypersurfaces in affine space, including graphs of meromorphic functions.

Findings

Hyperplane arrangement complements are Oka under certain conditions.

Some hypersurfaces in affine space have Oka complements.

Results relate Oka properties to the geometry of hypersurfaces and meromorphic functions.

Abstract

Oka manifolds can be viewed as the "opposite" of Kobayashi hyperbolic manifolds. Kobayashi asked whether the complement in projective space of a generic hypersurface of sufficiently high degree is hyperbolic. Therefore it is natural to investigate Oka properties of complements of low degree hypersurfaces. We determine which complements of hyperplane arrangements in projective space are Oka. A related question is which hypersurfaces in affine space have Oka complements. We give some results for graphs of meromorphic functions.