  We study the hyperbolicity of the log variety $(\mathbb{P}^n, X)$, where $X$
is a very general hypersurface of degree $d\geq 2n+1$ (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.
