Differential Equations on Complex Projective Hypersurfaces of Low Dimension

TL;DR

This paper establishes degree bounds for smooth complex projective hypersurfaces in projective space that guarantee all entire curves satisfy algebraic differential equations of a certain order, advancing understanding of hyperbolicity.

Contribution

It provides effective lower bounds on the degree of hypersurfaces ensuring algebraic differential equations of order equal to the dimension, and shows the non-existence of such equations for lower orders.

Findings

Every entire curve in the hypersurface satisfies an algebraic differential equation of order n.

No algebraic differential equations of order less than n exist for such hypersurfaces.

Explicit degree bounds are given for the existence of these differential equations.

Abstract

Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of $\mathbb P^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic differential equation of order $k=n=\dim X$, and also similar bounds for order $k>n$. Moreover, for every integer $n\ge 2$, we show that there are no such algebraic differential equations of order $k<n$ for a smooth hypersurface in $\mathbb P^{n+1}$.