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 X^n−1 in P^n(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 > (sqrt(n) log n)^n,
and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain:
d > (n log n)^n.
The latter improves d > n^2n obtained by Merker in arxiv.org/1807/11309/.
Admitting a certain technical conjecture I0 > I0',
the method employed (Diverio-Merker-Rousseau, Bérczi, Darondeau)
conducts to constant power n, namely to:
d > 2^5n
and, respectively, to:
d > 4^5n.
In Spring 2021, a forthcoming prepublication based on intensive computer explorations
will present several subconjectures supporting the belief that I0 > I0',
a conjecture which will be established up to dimension n = 50.
Domains
Mathematics [math]
Origin : Files produced by the author(s)