Bombieri–Lang conjecture

Let $K$ be a number field and let $X$ be a smooth projective geometrically integral variety over $K$ of dimension $d\ge1$. Let $\omega_X$ be its canonical bundle. $X$ is of general type if $\omega_X$ is big, that is,

$$\dim_K H^0\bigl(X,\omega_X^{\otimes m}\bigr)\ \ge\ c\,m^{d}$$

for some $c>0$ and all sufficiently large and divisible $m$. A curve is of general type exactly when its genus is at least $2$. Write $X(K)$ for the set of $K$-rational points of $X$.

Conjecture (Bombieri–Lang). If $X$ is of general type, then $X(K)$ is not Zariski dense in $X$. That is, there is a proper Zariski-closed subset $Z\subsetneq X$ with $X(K)\subseteq Z(K)$.

A solution must prove this for every number field $K$ and every such $X$. Non-density is a birational property, so the conjecture for smooth projective varieties implies the same statement for any integral projective variety whose resolution of singularities is of general type. The case $d=1$ is Faltings' theorem. The conjecture is open already for surfaces.

Lang also proposed a stronger form [1]. There is a proper Zariski-closed subset $Z\subsetneq X_{\bar K}$, depending only on the geometry of $X$, such that $(X\setminus Z)(L)$ is finite for every finite extension $L/K$. The problem here is the non-density statement above.

Background

Bombieri proposed the conjecture for surfaces around 1980, and Lang formulated it in all dimensions [1]. It generalizes Mordell's conjecture from curves to higher dimensions. The conclusion for $d\ge2$ is non-density, since a surface of general type can contain infinitely many rational points on a curve of genus $0$ or $1$.

The conjecture is part of a broader dictionary between arithmetic and complex geometry. Lang conjectured that the exceptional set $Z$ in the strong form can be taken to be the Zariski closure of the images of all nonconstant rational maps from abelian varieties and from $\mathbb P^1$ to $X$. He related this to Kobayashi hyperbolicity in complex geometry [1]. Vojta's height conjecture, an arithmetic analogue of Nevanlinna's second main theorem, implies Bombieri–Lang [3]. Caporaso, Harris and Mazur showed that Bombieri–Lang implies a uniform bound for the number of rational points on curves of genus $g$ over $K$, depending only on $g$ and $K$ [6].