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

TL;DR

This paper proves that for high-degree generic projective hypersurfaces, all entire holomorphic maps from the complex plane are contained in a proper subvariety of codimension at least two, confirming a special case of the Kobayashi conjecture.

Contribution

It establishes a new effective bound on the degree of hypersurfaces ensuring entire curves are contained in a proper subvariety, confirming the Kobayashi conjecture for threefolds in projective 4-space.

Findings

Existence of a proper subvariety Y containing all entire curves

Effective degree bound for hypersurfaces: at least 2^{n^5}

Confirmation of Kobayashi conjecture for threefolds in P^4

Abstract

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