Hyperbolicity and GCD for n+1 divisors with non-empty intersection

1. Problem Statement

The paper addresses the Green-Griffiths-Lang (GGL) Conjecture, which predicts that entire curves on quasi-projective varieties of log-general type are algebraically degenerate (i.e., their images lie in a proper algebraic subvariety).

Specifically, the authors focus on a geometric setting where:

• $X$ is a complex projective variety of dimension $n$.
• The boundary divisor $D$ is the union of $n+1$ effective Cartier divisors $D_1, \dots, D_{n+1}$.
• These divisors are numerically parallel (there exist positive integers $a_i$ such that $a_i D_i$ are all numerically equivalent to an ample divisor $D$).
• Crucial Novelty: Unlike previous results that often assume the divisors have empty intersection (normal crossings), this paper allows the divisors to have non-empty intersection (specifically, $\bigcap_{i=1}^{n+1} D_i \neq \emptyset$).

The core challenge is to prove that under specific local conditions at the intersection points, any non-constant holomorphic map $f: \mathbb{C} \to X \setminus \bigcup D_i$ must be algebraically degenerate. Additionally, the paper seeks to establish a GCD-type estimate (bounding the common zeros of sections) in this higher-dimensional, intersecting setting.

2. Methodology

The authors extend the techniques developed by Levin, Huang, and Xiao (specifically their work on surfaces in [5]) to higher dimensions ($n \geq 2$). The methodology relies on Nevanlinna theory and Diophantine approximation tools.

A. Key Definitions and Tools

• $\beta$-constant: A generalization of Autissier's $\alpha$-constant, defined for a line sheaf $L$ and a closed subscheme $Y$ as:
  $$\beta(L, Y) = \liminf_{N \to \infty} \frac{\sum_{m \geq 1} h^0(L^N \otimes \mathcal{I}_Y^m)}{N h^0(L^N)}$$
  This measures the "density" of sections vanishing along $Y$.
• Regular Sequences of Subschemes: The authors generalize the concept of divisors intersecting properly to chains of closed subschemes $Y_1 \supset Y_2 \supset \dots \supset Y_m$. This allows them to handle the local geometry near intersection points where multiple divisors meet.
• Moving Filtrations: Instead of fixed filtrations used in lower-dimensional cases, they utilize moving filtrations (introduced by Autissier) on the spaces of global sections $H^0(X, L^N)$. This allows for a more flexible adaptation to the specific geometry of the intersection points.

B. Proof Strategy

1. Second Main Theorem for Regular Sequences (Theorem 1.9): The authors first prove a generalized Second Main Theorem (SMT) for a collection of divisors and closed subschemes. This theorem provides an inequality involving proximity functions ($m_f$) and characteristic functions ($T_{L,f}$) weighted by $\beta$-constants.
   • Technique: They construct specific filtrations on $H^0(X, L^N)$ adapted to the divisors and their intersections. By analyzing the least upper bound of divisors associated with bases adapted to these filtrations, they derive the necessary inequality.
2. Convex Combination Argument: To handle the $n+1$ divisors with non-empty intersection, they employ a convex combination strategy. They apply the generalized SMT to different subsets of $n$ divisors, weighting the resulting inequalities to isolate the contribution of the intersection locus.
3. Blow-up Technique: In the proof of the main hyperbolicity theorem, they utilize a blow-up $\pi: \tilde{X} \to X$ at an intersection point $Q$. This separates the divisors, allowing the application of the SMT on the blown-up space where the exceptional divisor $E$ plays a critical role in controlling the local behavior.

3. Key Contributions and Results

A. Main Hyperbolicity Theorem (Theorem 1.5)

The paper proves that if $D_1, \dots, D_{n+1}$ are numerically parallel effective divisors on an $n$-dimensional variety $X$ with non-empty intersection, and they satisfy specific local conditions, then all entire curves are algebraically degenerate.

Conditions:

1. Any $n$-tuple of divisors intersects properly.
2. For every point $Q$ in the total intersection $\bigcap D_i$ and every subset $I$ of size $n$, the following inequality holds:
   $$\beta\left(D, \left(\bigcap_{i \in I} a_i D_i\right)_Q\right) > 1$$
   (Here, the subscript $Q$ denotes the scheme supported at $Q$).

Result: There exists a proper Zariski-closed subset $Z \subset X$ such that the image of any non-constant holomorphic map $f: \mathbb{C} \to X \setminus \bigcup D_i$ is contained in $Z$.

B. GCD-Type Estimate (Theorem 1.10)

The authors derive a GCD estimate, which bounds the proximity function of the intersection of $n$ divisors relative to the characteristic function of the ample divisor.

• Result: If the intersection of a specific subset of $n$ divisors contains more than one point, then for any $\epsilon > 0$, there exists a proper Zariski-closed set $Z$ such that for $f \not\subset Z$:
  $$T_{\bigcap_{i \in I_0} a_i D_i, f}(r) \leq \epsilon T_{D, f}(r)$$
  This implies that the curve cannot stay "too close" to the intersection of the divisors unless it is contained in a specific algebraic subvariety.

C. Generalized Second Main Theorem (Theorem 1.9)

A significant theoretical contribution is the extension of the Ru-Vojta type SMT to regular sequences of closed subschemes. This theorem (Equation 1.3) handles the maximum of weighted sums of proximity functions for divisors and their intersections, providing a robust tool for higher-dimensional hyperbolicity problems.

4. Significance and Implications

1. Resolution of Non-Empty Intersection Cases: Prior to this work, results on $n+1$ numerically parallel divisors largely required the divisors to be in general position (empty intersection). This paper successfully tackles the much more difficult case where divisors intersect, a scenario that introduces significant analytic challenges in controlling local behavior.
2. Higher-Dimensional Generalization: The paper successfully lifts the surface-level results of Levin-Huang-Xiao [5] to arbitrary dimension $n$. This is non-trivial due to the complexity of the geometry of intersections in higher dimensions.
3. Sharpness of Conditions: The authors provide examples (Section 4) demonstrating that the condition $\beta > 1$ is sharp. If the $\beta$-constant condition is violated, one can construct entire curves that are algebraically non-degenerate, proving the necessity of their hypotheses.
4. Bridge to Arithmetic Geometry: The methods and results have direct analogues in Diophantine approximation over number fields (the Lang-Vojta conjecture). The paper notes that their results extend to the number field setting, offering new tools to study integral points on varieties with intersecting divisors, a case previously open even for surfaces ($n=2$).
5. Refinement of Nevanlinna Theory: The introduction of filtrations adapted to regular sequences of subschemes and the precise calculation of $\beta$-constants for intersections provide a refined toolkit for future research in value distribution theory.

In summary, this paper represents a major advancement in the study of hyperbolicity for quasi-projective varieties, successfully overcoming the technical barrier of non-empty divisor intersections in higher dimensions and providing a unified framework for both complex analytic and arithmetic applications.