Griffiths–Lang conjecture for holomorphic curves and simple normal crossing divisors

Let $X$ be a nonsingular complex projective algebraic variety, let $D$ be a simple normal crossing divisor on $X$, and let $m_{\mathbf{x},D}(r)$ and $T_{\mathbf{x},K_X^{-1}}(r)$ denote the proximity and order functions associated with a holomorphic curve $\mathbf{x}:\mathbb{C}\to X$. For an ample line bundle $E$ over $X$, set

$$S_{\mathbf{x}}(r)\coloneqq O\bigl(\log r+\log^{+}T_{\mathbf{x},E}(r)\bigr).$$

Griffiths–Lang conjecture. There exists a proper Zariski closed subset $Z_D\subsetneq X$ such that, for all holomorphic curves $\mathbf{x}:\mathbb{C}\to X$ with $\mathbf{x}(\mathbb{C})\nsubseteq Z_D$, and for any ample line bundle $E$ over $X$,

$$N_{\mathbf{x},\mathrm{Ram}}(r)+m_{\mathbf{x},D}(r)\leq T_{\mathbf{x},K_X^{-1}}(r)+S_{\mathbf{x}}(r),$$

where $K_X^{-1}$ is the anticanonical bundle of $X$. Giving an appropriate definition of the ramification counting function $N_{\mathbf{x},\mathrm{Ram}}(r)$ is part of the content of the conjecture. This is an important conjectural second main theorem in Nevanlinna theory; the source gives no evidence of a resolution, and the general ramification term remains part of the conjecture.