Lang–Vojta conjecture for varieties of general type

Let $X$ be a projective variety of general type and let $L$ be an ample line bundle. Then there is a proper algebraic subset $Z\subsetneq X$ and a constant $\alpha$ such that, for every smooth projective connected curve $C$ and every morphism $f:C\to X$ with $f(C)\not\subset Z$, one has

$$\deg f^*L\leq\alpha(2g(C)-2).$$

Lang–Vojta conjecture. The stated degree inequality holds uniformly for all such curves and morphisms outside the proper exceptional subset $Z$.

This is a particular form of the Lang–Vojta conjecture; the paper establishes it in the setting needed for its results on geometric specialness. The general conjecture remains open.