Degrees $d⩾(\sqrt{n}\log n{)}^n$ and $d⩾(n\log n{)}^n$ in the Conjectures of Green-Griffiths and of Kobayashi

Abstract

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 $d⩾2n$. For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain: \[ d \,\geqslant\, \big(\sqrt{n}\,{\sf log}\,n\big)^n, \] and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain: \[ d \,\geqslant\, \big(n\,{\sf log}\,n\big)^n. \] The latter improves $d⩾n^{2n}$ obtained by Merker in arxiv.org/1807/11309/. Admitting a certain technical conjecture $I_0⩾{\tilde{I}}_0$, the method employed (Diverio-Merker-Rousseau, Bérczi, Darondeau) conducts to constant power $n$, namely to: \[ d\ ,\geqslant\, 2^{5n} \qquad \text{and, respectively, to:} \qquad d \,\geqslant\, 4^{5n}. \] In Spring 2019, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_0⩾{\tilde{I}}_0$, a conjecture which will be established up to dimension $n=50$.

52 pages. Proofs completed without any computer help in this text, and relying heavily on Darondeau's Ph.D. (Orsay University, Paris, France, July 2014)