Nevanlinna and algebraic hyperbolicity

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair [Formula: see text], where [Formula: see text] is a projective variety and [Formula: see text] is an effective Cartier divisor on [Formula: see text]. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Păun and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596 ].