On the logarithmic Kobayashi conjecture

Abstract

We study the hyperbolicity of the log variety (ℙn,X), where X is a very general hypersurface of degree d ≧ 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 (ℙn, X) is of log-general type, give a new proof of the algebraic hyperbolicity of (ℙn, 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.