[Paper Review] Degrees $d \geqslant \big( \sqrt{n}\, \log\, n\big)^n$ and $d \geqslant \big( n\, \log\, n\big)^n$ in the Conjectures of Green-Griffiths and of Kobayashi

Joël Merker, The-Anh Ta|arXiv (Cornell University)|Jan 13, 2019

Analytic Number Theory Research10 references4 citations

TL;DR

This paper improves degree bounds for the Green-Griffiths and Kobayashi conjectures on hyperbolicity of generic hypersurfaces in complex projective space. By refining the Diverio-Merker-Rousseau and Darondeau framework with advanced estimates on jet differentials and a key technical conjecture $I_0 /geq ilde{I}_0$, it establishes that algebraic hypersurfaces of degree $d /geq ( ext{log} olimits n imes ext{sqrt}(n))^n$ are algebraically degenerate, and those of degree $d /geq (n ext{log} olimits n)^n$ are Kobayashi hyperbolic, significantly improving prior bounds.

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 $d \geqslant 2n$. 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 $d \geqslant n^{2n}$ 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: \[ d\ ,\geqslant\, 2^{5n} \qquad ext{and, respectively, to:} \qquad d \,\geqslant\, 4^{5n}. \] In Spring 2019, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_0 \geqslant \widetilde{I}_0$, a conjecture which will be established up to dimension $n = 50$.

Motivation & Objective

• To improve the lower degree bounds for generic hypersurfaces in $\mathbb{P}^n(\mathbb{C})$ to satisfy the Green-Griffiths and Kobayashi conjectures.
• To reduce the degree threshold for algebraic degeneracy of entire holomorphic curves in the complement of a hypersurface.
• To establish that the conjecture $I_0 \geq \tilde{I}_0$ implies significantly tighter degree bounds, approaching the 'celestial' horizon near $d \geq 2n$.
• To provide effective, explicit degree thresholds for hyperbolicity and algebraic degeneracy in both the complement and compact cases.
• To support the conjecture $I_0 \geq \tilde{I}_0$ via computer explorations up to dimension $n=50$, laying groundwork for future proof.

Proposed method

• Applies the Diverio-Merker-Rousseau method of jet differentials and holomorphic Morse inequalities to control the growth of jet sections.
• Uses a rational function $C(t_1,\dots,t_n)$ encoding jet order $\kappa = n$ and reinterprets it as a convergent power series in new variables $w_2,\dots,w_n$.
• Employs a key technical conjecture $I_0 \geq \tilde{I}_0$, equivalent to Problem 4.2, to strengthen estimates on the positivity of jet differential forms.
• Performs detailed asymptotic and trigonometric analysis on functions $h_{\ell,\rho}(\theta)$ to prove positivity in critical angular intervals.
• Applies the classical inequality $|\sin \gamma| \geq \frac{1}{2}|\gamma|$ for $|\gamma| \leq \frac{\pi}{2}$ to bound trigonometric terms in the positivity proof.
• Adapts Darondeau’s work on jet differentials to extend results from the complement case to the compact case (hypersurface itself).

Experimental results

Research questions

• RQ1 Can the degree bound for algebraic degeneracy of entire holomorphic curves in the complement of a generic hypersurface in $\mathbb{P}^n(\mathbb{C})$ be improved below $d \geq (5n)^2 n^n$?
• RQ2 What is the minimal degree $d$ such that a generic hypersurface $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ is Kobayashi hyperbolic?
• RQ3 Does the technical conjecture $I_0 \geq \tilde{I}_0$ lead to a substantial improvement in the degree threshold for hyperbolicity?
• RQ4 Can the positivity of the jet differential form be established uniformly across all dimensions $n$ using the current method?
• RQ5 To what extent can computer explorations support the validity of $I_0 \geq \tilde{I}_0$ up to dimension $n=50$?

Key findings

• For generic hypersurfaces of degree $d \geq (\sqrt{n} \log n)^n$, all nonconstant entire holomorphic curves in the complement $\mathbb{P}^n \setminus \mathbb{X}^{n-1}$ are algebraically degenerate and contained in a proper subvariety of codimension $\geq 2$.
• For the same degree bound, all nonconstant entire holomorphic curves on the hypersurface $\mathbb{X}^{n-1}$ are also algebraically degenerate and contained in a proper subvariety of codimension $\geq 2$.
• For Kobayashi hyperbolicity, the paper improves the bound from $d \geq n^{2n}$ to $d \geq (n \log n)^n$, establishing that $\mathbb{P}^n \setminus \mathbb{X}^{n-1}$ is Kobayashi hyperbolic and $\mathbb{X}^{n-1}$ is hyperbolic for such degrees.
• Assuming the conjecture $I_0 \geq \tilde{I}_0$, the degree bound improves further to $d \geq 2^{5n}$ for algebraic degeneracy and $d \geq 4^{5n}$ for Kobayashi hyperbolicity.
• The positivity of the key function $h_{\ell,\rho}(\theta)$ is proven for all $\ell \geq 3$ and $0 \leq \rho \leq 0.25$ in the interval $[7\pi/(4\ell), \pi]$, which is essential for the main estimates.
• The conjecture $I_0 \geq \tilde{I}_0$ is supported by intensive computer explorations, with verification up to dimension $n=50$ expected in a forthcoming preprint.