Vojta's conjecture

Let ${\displaystyle F}$ be a number field, let ${\displaystyle X/F}$ be a non-singular algebraic variety, let ${\displaystyle D}$ be an effective divisor on ${\displaystyle X}$ with at worst normal crossings, let ${\displaystyle H}$ be an ample divisor on ${\displaystyle X}$, and let ${\displaystyle K_{X}}$ be a canonical divisor on ${\displaystyle X}$. Choose Weil height functions ${\displaystyle h_{H}}$ and ${\displaystyle h_{K_{X}}}$ and, for each absolute value ${\displaystyle v}$ on ${\displaystyle F}$, a local height function ${\displaystyle \lambda _{D,v}}$. Fix a finite set of absolute values ${\displaystyle S}$ of ${\displaystyle F}$, and let ${\displaystyle \epsilon >0}$. Then there is a constant ${\displaystyle C}$ and a non-empty Zariski open set ${\displaystyle U\subseteq X}$, depending on all of the above choices, such that

${\displaystyle \sum _{v\in S}\lambda _{D,v}(P)+h_{K_{X}}(P)\leq \epsilon h_{H}(P)+C\quad {\hbox{for all }}P\in U(F).}$

Examples:

1. Let ${\displaystyle X=\mathbb {P} ^{N}}$. Then ${\displaystyle K_{X}\sim -(N+1)H}$, so Vojta's conjecture reads ${\displaystyle \sum _{v\in S}\lambda _{D,v}(P)\leq (N+1+\epsilon )h_{H}(P)+C}$ for all ${\displaystyle P\in U(F)}$.
2. Let ${\displaystyle X}$ be a variety with trivial canonical bundle, for example, an abelian variety, a K3 surface or a Calabi-Yau variety. Vojta's conjecture predicts that if ${\displaystyle D}$ is an effective ample normal crossings divisor, then the ${\displaystyle S}$-integral points on the affine variety ${\displaystyle X\setminus D}$ are not Zariski dense. For abelian varieties, this was conjectured by Lang and proven by Faltings (1991).
3. Let ${\displaystyle X}$ be a variety of general type, i.e., ${\displaystyle K_{X}}$ is ample on some non-empty Zariski open subset of ${\displaystyle X}$. Then taking ${\displaystyle S=\emptyset }$, Vojta's conjecture predicts that ${\displaystyle X(F)}$ is not Zariski dense in ${\displaystyle X}$. This last statement for varieties of general type is the Bombieri–Lang conjecture.

There are generalizations in which ${\displaystyle P}$ is allowed to vary over ${\displaystyle X({\overline {F}})}$, and there is an additional term in the upper bound that depends on the discriminant of the field extension ${\displaystyle F(P)/F}$.

There are generalizations in which the non-archimedean local heights ${\displaystyle \lambda _{D,v}}$ are replaced by truncated local heights, which are local heights in which multiplicities are ignored. These versions of Vojta's conjecture provide natural higher-dimensional analogues of the ABC conjecture.

• Vojta, Paul (1987). Diophantine approximations and value distribution theory. Lecture Notes in Mathematics. Vol. 1239. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0072989. ISBN 978-3-540-17551-3. MR 0883451. Zbl 0609.14011.
• Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Annals of Mathematics. 123 (3): 549–576. doi:10.2307/2944319. MR 1109353.