[Paper Review] Degeneracy of entire curves into higher dimensional complex manifolds

Ya Deng|arXiv (Cornell University)|Mar 7, 2016

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper generalizes McQuillan's approach to the Green-Griffiths conjecture by studying entire curves tangent to holomorphic foliations on higher-dimensional complex manifolds. It introduces the concept of weakly reduced singularities for foliations, enabling a refined analysis of Ahlfors currents and their intersection with tangent and normal bundles, ultimately leading to a strategy for proving algebraic degeneracy of entire curves on surfaces of general type.

ABSTRACT

Pursuing McQuillan's philosophy in proving the Green-Griffiths conjecture for certain surfaces of general type, we deal with the algebraic degeneracy of entire curves tangent to holomorphic foliations by curves. Inspired by the recent work of Paun and Sibony, we study the intersection of Ahlfors current with the tangent bundle of the foliation, and derive some consequences. In particular, we introduce the definition of weakly reduced singularities for foliations by curves, which requires less work than the exact classification for foliations. Finally we discuss the strategy to prove the Green-Griffiths conjecture for complex surfaces.

Motivation & Objective

• To extend McQuillan's strategy for proving the Green-Griffiths conjecture to higher-dimensional complex manifolds.
• To define and utilize the notion of weakly reduced singularities for 1-dimensional foliations, reducing the need for full classification of singularities.
• To establish a refined intersection formula for Ahlfors currents with the tangent bundle of a foliation.
• To provide a new pathway toward proving the Green-Griffiths conjecture by combining current theory with birational geometry and foliation resolution.
• To formulate and support conjectures on singularity reduction and generalization of Brunella's theorem to higher dimensions.

Proposed method

• Uses Ahlfors currents $T[f]$ associated with entire curves $f: \mathbb{C} \to X$ tangent to a foliation $\mathcal{F}$, treating them as $(1,1)$-cohomology classes.
• Applies Siu's refined tautological inequality to analyze the intersection $\langle T[f], c_1(T_{\mathcal{F}}) \rangle + T(f, \mathcal{J}_{\mathcal{F}})$, where $\mathcal{J}_{\mathcal{F}}$ encodes singularities.
• Introduces the error term $T(f, \mathcal{J}_{\mathcal{F}})$ as a non-negative real number representing the interaction between the current and the singular locus.
• Employs birational transformations, including finite sequences of blow-ups, to resolve singularities into weakly reduced ones.
• Leverages the Demailly-Semple tower construction to analyze the Zariski closure of lifts of entire curves and their behavior under projection.
• Relies on conjectural generalizations of Brunella's non-negativity result for normal bundle intersections to higher dimensions.

Experimental results

Research questions

• RQ1 Can the algebraic degeneracy of entire curves on complex surfaces of general type be established via foliation-theoretic methods?
• RQ2 To what extent can the singularity classification of 1-dimensional foliations be simplified using the concept of weakly reduced singularities?
• RQ3 Does the intersection $T[f] \cdot c_1(N_{\mathcal{F}})$ remain non-negative under generalized conditions, even when singularities are not isolated?
• RQ4 Can the strategy of resolving foliations into weakly reduced singularities be made algorithmic or universally applicable?
• RQ5 Can Brunella's theorem on non-negative normal bundle intersection be extended to higher-dimensional manifolds with foliations?

Key findings

• The paper establishes a refined formula for $\langle T[f], c_1(T_{\mathcal{F}}) \rangle + T(f, \mathcal{J}_{\mathcal{F}}) \geq 0$, which improves upon the result in [PS14].
• For surfaces, the inequality $T[f] \cdot T_{\mathcal{F}} \geq 0$ holds under the assumption of Zariski density and non-degeneracy.
• If $\mathcal{F}$ has absolutely isolated singularities and $K_{\mathcal{F}}$ is big, then all entire curves tangent to $\mathcal{F}$ are algebraically degenerate.
• The introduction of weakly reduced singularities allows control over the error term $T(f, \mathcal{J}_{\mathcal{F}})$ without requiring full classification of singularities.
• Under the conjecture that singularities can be resolved into weakly reduced ones, and assuming a generalized Brunella theorem, the Green-Griffiths conjecture for surfaces of general type follows.
• The contradiction arises when $\langle T[\hat{g}], c_1(N_{\hat{\mathcal{F}}}) \rangle < 0$ contradicts the expected non-negativity from Conjecture 3.2, implying algebraic degeneracy.