On the logarithmic Kobayashi conjecture

Abstract

We study the hyperbolicity of the log variety $({\mathbb{P}}^n,X)$, where $X$ is a very general hypersurface of degree $d\ge 2n+1$ (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of $({\mathbb{P}}^n,X)$ is of log-general type, give a new proof of the algebraic hyperbolicity of $({\mathbb{P}}^n,X)$, and exclude the existence of maximal rank families of entire curves in the complement of the universal degree $d$ hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.

17 pages