The generalized Green–Griffiths–Lang conjecture for smooth quasi-projective varieties

Let $X$ be a smooth quasi-projective variety. Define the special loci

$$\mathrm{Sp}_{\mathrm{sab}}(X),\quad \mathrm{Sp}_{\mathrm{h}}(X),\quad \mathrm{Sp}_{\mathrm{alg}}(X),\quad \mathrm{Sp}_{\mathrm{p}}(X)$$

as follows: $\mathrm{Sp}_{\mathrm{sab}}(X)$ is the Zariski closure of the union of images of non-constant rational maps from non-trivial semi-abelian varieties that are regular outside a subset of codimension at least two; $\mathrm{Sp}_{\mathrm{h}}(X)$ is the Zariski closure of the union of images of non-constant holomorphic maps from $\mathbb{C}$; $\mathrm{Sp}_{\mathrm{alg}}(X)$ is the Zariski closure of the union of positive-dimensional closed subvarieties that are not of log general type; and $\mathrm{Sp}_{\mathrm{p}}(X)$ is the Zariski closure of the union of images of holomorphic maps from $\mathbb{D}^*$ with an essential singularity at the origin. Generalized Green–Griffiths–Lang conjecture. The following properties are equivalent: $X$ is of log general type; $X$ is strongly of general type, meaning $\mathrm{Sp}_{\mathrm{alg}}(X)\subsetneqq X$; $X$ is pseudo Picard hyperbolic, meaning $\mathrm{Sp}_{\mathrm{p}}(X)\subsetneqq X$; $X$ is pseudo Brody hyperbolic, meaning $\mathrm{Sp}_{\mathrm{h}}(X)\subsetneqq X$; and $\mathrm{Sp}_{\mathrm{sab}}(X)\subsetneqq X$. This extends the Green–Griffiths–Lang principle from projective varieties to the quasi-projective setting and relates log general type to several algebraic and analytic notions of non-hyperbolicity. The paper establishes the conjecture for varieties of log-general type and maximal quasi-Albanese dimension, but the equivalence in general remains open.