[Paper Review] Hyperbolicity of Generic High-Degree Hypersurfaces in Complex Projective Space

Yum-Tong Siu|arXiv (Cornell University)|Sep 12, 2012

Meromorphic and Entire Functions16 references4 citations

TL;DR

This paper establishes the hyperbolicity of generic high-degree hypersurfaces in complex projective space by constructing holomorphic jet differentials with controlled vanishing properties using slanted vector fields and meromorphic jet differentials of low pole order. The key result is that for degrees exceeding an explicit threshold depending on dimension, there are no nonconstant entire holomorphic maps from the complex line to such hypersurfaces or their complements.

ABSTRACT

We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) nonexistence of nontrivial sets of entire functions satisfying certain polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets.

Motivation & Objective

• To establish the hyperbolicity of generic hypersurfaces of sufficiently high degree in complex projective space of dimension n ≥ 3.
• To extend this result to the complement of such hypersurfaces in CP^n for n ≥ 2.
• To develop a method based on slanted vector fields and jet differentials to prove nonexistence of entire curves.
• To generalize the approach to include slowly varying coefficients and moving targets, yielding Second Main Theorems for jet differentials.
• To provide a constructive proof of hyperbolicity using linear algebra and differentiation techniques inspired by Bloch’s 1926 work.

Proposed method

• Constructs slanted vector fields in the vertical jet space of the universal hypersurface to control the behavior of jet differentials.
• Uses meromorphic jet differentials of low pole order (magnitude δ^{1−ε}) on CP^n and pulls them back to the hypersurface X.
• Applies injectivity of the pullback map for jet differentials (Lemma 3.4) to ensure sufficient linear independence of pullbacks.
• Forms nontrivial holomorphic jet differentials on X as C-linear combinations of pullbacks that vanish on an ample divisor of X (Proposition 3.8).
• Employs the technique of slanted vector fields to derive vanishing identities that force entire maps to be constant.
• Applies Nevanlinna theory and Second Main Theorem arguments for log-pole jet differentials with slowly moving targets.

Experimental results

Research questions

• RQ1 For which degrees δ ≥ δ_n is a generic hypersurface X ⊂ ℙ_n hyperbolic, i.e., admits no nonconstant holomorphic map from ℂ?
• RQ2 Can the hyperbolicity of the complement ℙ_n − X be established for generic hypersurfaces of sufficiently high degree?
• RQ3 How can jet differentials be constructed explicitly on high-degree hypersurfaces using pullbacks of meromorphic jet differentials from ℙ_n?
• RQ4 What role do slanted vector fields play in generating vanishing identities for jet differentials that obstruct entire curves?
• RQ5 Can Second Main Theorems for jet differentials be formulated in the context of slowly moving targets and varying coefficients?

Key findings

• For each n ≥ 3, there exists an explicit threshold δ_n such that any generic hypersurface in ℙ_n of degree δ ≥ δ_n is hyperbolic, meaning it admits no nonconstant holomorphic map from ℂ.
• For each n ≥ 2, there exists an explicit threshold δ_n^* such that the complement of a generic hypersurface of degree δ ≥ δ_n^* in ℙ_n is also hyperbolic.
• The construction of holomorphic jet differentials on X relies on pulling back meromorphic jet differentials of low pole order from ℙ_n, with injectivity of the pullback map ensuring sufficient linear independence.
• A nontrivial holomorphic jet differential on X is obtained as a C-linear combination of pullbacks that vanishes on an ample divisor of X, enabling the use of vanishing arguments.
• The method generalizes to Second Main Theorems for jet differentials with slowly moving targets, where the proximity function is bounded by the counting function up to lower-order terms.
• The approach recovers Cartan’s Second Main Theorem for hyperplanes in general position as a special case via the Wronskian construction of jet differentials with controlled poles.