An algebraic variety $V$ is said to be of *general type* if it is of maximal Kodaira dimension. If $V$ is defined over a number field $K$, then one has the following conjecture due to Lang (Bombieri had made a similar conjecture in the case of surfaces; thus this conjecture is also known as the Bombieri-Lang conjecture):


If $V$ is a variety of general type defined over a number field $K$, then the set of $K$-rational points $V(K)$ of $V$ is not Zariski dense.


When $\dim V = 1$, this is Mordell's conjecture and was famously solved by Faltings in the 1980s.


Has there been any progress towards Lang's conjecture, in any situation beyond the curve case, since then?


"Progress" does not necessarily mean a proof, but even an outline of what a possible proof might look like or a program towards a proof would suffice. For example, Faltings, in his proof of Mordell's conjecture, had carried out a program that was already known at the time: given Parshin's trick, one shows that Mordell's conjecture is implied by Shafarevich's conjecture, and this is what Faltings actually proves. Is there anything like this beyond the curve case?



Lang's conjecture beyond the curve case
Faltings' second proof extends to subvarieties of abelian varieties $A$.
(The exceptional locus consists of the translates of abelian subvarieties
of $A$.)
That's a very special case, but it includes varieties birational with
symmetric powers of curves: as long as $d$ is less than the genus of $C$,
the $d$-th symmetric product ${\rm Sym}^d C$ is birational with its image in
${\rm Pic}^d(C)$. Rational points of ${\rm Sym}^d C$ come from
points of $C$ defined over a varying field extension $K'/K$
of degree at most $d$.


Besides that, if $V$ has an unramified cover $V'$ then one can do a
"descent" to find a finite extension $K'/K$ such that $V(K)$ lifts to $V'(K')$.
It may be that $V'$ birationally embeds into an abelian variety even though
$V$ did not; for example, if $V$ is the quotient of $C \times C'$ by
a finite group acting freely on the product of curves $C,C'$
so that the fixed subspace of $H^1$ is trivial. In particular
such varieties satisfy the Bombieri-Lang conjecture.
But that, too, is a very special case. There might not be any others known yet.


