A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree

Abstract

We show that for every smooth generic projective hypersurface $X\subset {\mathbb{P}}^{n+1}$, there exists a proper subvariety $Y⊊X$ such that ${\mathrm{codim}}_{X}Y\ge 2$ and for every non constant holomorphic entire map $f:\mathbb{C}\to X$ one has $f(\mathbb{C})\subset Y$, provided $°X\ge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in ${\mathbb{P}}^4$.

arXiv comment7 pages, no figures, comments are welcome. Corrected typos, added a small section with an open question, references updated. Final version, to appear on J. Reine Angew. Math.