On the logarithmic Kobayashi conjecture

Details On the logarithmic Kobayashi conjecture On the logarithmic Kobayashi conjecture

2006-03-30

arxiv.org/abs/math/0603712v1

Mathematics

Algebraic Geometry

17 pages

Scientific paper

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.

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