Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $\mathbb {X}^{n-1} \subset \mathbb {P}^{n}(\mathbb {C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the ‘celestial’ horizon lies near $d\geqslant 2n$.

For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain $$d\geqslant (\sqrt{n}\log n)^n,$$ and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain $$ d\geqslant \left (n\log n\right )^{n}. $$ The latter improves $d \geqslant n^{2n}$ obtained by Merker in arxiv.org/1807/11309/.

Admitting a certain technical conjecture $I_{0} \geqslant \widetilde {I}_{0} $, the method employed (Diverio-Merker-Rousseau, Bérczi, Darondeau) conducts to constant power $n$, namely to $$ d\geqslant 2^{5n}\quad \text {and, respectively, to}\quad d\geqslant 4^{5n}. $$

In Spring 2021, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_{0} \geqslant \widetilde {I}_{0}$ a conjecture which will be established up to dimension $n=50$.