Generalized Green-Griffiths-Lang Conjecture

• The Generalized Green-Griffiths-Lang Conjecture is a central idea in algebraic geometry that relates the positivity of a variety to the confinement of its entire holomorphic curves.
• It extends the classical conjecture by incorporating directed varieties, strong hyperbolicity via special loci, and relative settings through advanced jet bundle, Morse inequalities, and Nevanlinna theory.
• Recent progress has led to effective degree bounds and deeper analytic insights, while open problems persist in characterizing exceptional loci and exploring arithmetic and logarithmic analogues.

The Generalized Green-Griffiths-Lang (GGL) Conjecture is a central statement in modern complex algebraic geometry, predicting deep links between the positivity properties of a variety (specifically, being of general type) and the degeneracy and distribution of entire holomorphic curves within it. The conjecture has evolved to encompass various "generalized" forms: directed varieties, orbifold/logarithmic situations, arithmetic analogues, and, as introduced in recent work, relative or "family" contexts. Below is a detailed exposition of the classical and generalized conjectures, foundational definitions, key analytic and algebro-geometric methodologies, major results, and future directions.

1. Classical Green-Griffiths-Lang Conjecture

Let $X$ be a complex projective variety. An entire curve in $X$ is a nonconstant holomorphic map $f: \mathbb{C} \rightarrow X$. The exceptional locus (also called the Green-Griffiths locus) $\mathrm{Exc}(X) \subset X$ is defined as the Zariski closure of the union of all images of entire curves. The variety $X$ is said to be of general type if its canonical bundle (or a smooth birational model's) is big: $$\omega_X\ \text{is big} \ \Longleftrightarrow\ \limsup_{m\to\infty} \frac{\dim H^0(X,\omega_X^{\otimes m})}{m^{\dim X}} > 0.$$ Conjecture (Green-Griffiths-Lang):

If $X$ is of general type, $\mathrm{Exc}(X)$ is a proper Zariski-closed subset of $X$; equivalently, any entire curve must land in a proper subvariety of $X$.

2. Generalizations: Directed, Strong, and Relative Forms

2.1 Directed Varieties

A projective directed variety is a pair $(X, V)$ with $V \subset T_X$ a (possibly singular) linear subspace. An entire curve tangent to $V$ is $f: \mathbb{C} \to X$ with $df(T_\mathbb{C}) \subset V$. The canonical sheaf $K_V$ is appropriately defined via locally bounded sections or the reflexive hull of $\det V^*$. The generalized conjecture asserts: If $K_V$ is big, all $V$-tangent entire curves are contained in a proper subvariety of $X$ (Demailly, 2010, Demailly, 2014).

2.2 Strong GGL: The “Special Sets” and Hyperbolicity

Lang and collaborators introduced a strong form involving various loci:

• $\Sp(X)$: union of positive-dimensional closed subvarieties not of general type,
• $\Sp_{ab}(X)$: union of images of nonconstant rational maps from abelian varieties,
• $\Sp_h(X)$: union of all entire curves.

Strong GGL conjecture: For a projective variety $X$,

1. $\Sp(X) = \Sp_{ab}(X) = \Sp_h(X)$,
2. This locus is Zariski-closed,
3. $\Sp(X) \neq X$ if and only if $X$ is of general type.

The expectation is that for general type, these special loci are proper and coincide (Brunebarbe, 2022).

2.3 The Relative Green-Griffiths-Lang Conjecture

Given a proper morphism $f: X \to Y$ of complex projective varieties, the relative conjecture defines the analogs:

• $\Sp_{alg}(X/Y)$: union of positive-dimensional subvarieties of fibers $X_y$ not of general type,
• $\Sp_{ab}(X/Y)$: union of images of nonconstant rational maps from abelian varieties to fibers,
• $\Sp_h(X/Y)$: union of entire curves contained in fibers.

Relative GGL conjecture: For a proper morphism $X \to Y$, the following are equivalent: (i) $X \to Y$ is of general type (generic fiber), (ii) $\Sp_{alg}(X/Y)$ is not Zariski dense in $X$, (iii) $\Sp_{ab}(X/Y)$ is not Zariski dense in $X$, (iv) $\Sp_h(X/Y)$ is not Zariski dense in $X$ (Brunebarbe, 2023).

3. Main Techniques: Jet Bundles, Morse Inequalities, and Value Distribution

3.1 Jet Bundles and Jet Differentials

Jet bundle techniques are central:

• The construction of Green-Griffiths jet bundles $E_{k,m}^{GG}$ and the Semple towers, encoding the spaces of weighted algebraic differential operators vanishing on k-jets of entire curves (Demailly, 2010, Merker, 2010).
• Invariant jet differentials are defined via the action of the reparametrization group and are crucial for producing global sections that force entire curves to satisfy high-order algebraic differential equations (Berczi, 2010, Berczi, 2015).

3.2 Holomorphic Morse Inequalities

Morse inequalities are employed (often in their singular and analytic forms) to provide effective bounds on the dimensions of large tensor powers of line bundles over jet bundles, leading to global sections if the line bundle is big (Demailly, 2010, Cadorel, 27 Jun 2024). For example, the positivity of an intersection number

$A^{N} - N\,A^{N-1}\cdot B$

for nef $A, B$ on a suitable weighted projective bundle implies the existence of global jet differential sections with controlled vanishing (Cadorel, 27 Jun 2024).

3.3 Localization and Residue Techniques

In the context of generic hypersurfaces, equivariant localization on the Demailly–Semple tower (and its generalizations using Thom polynomials of Morin singularities) provides iterated-residue formulas for explicit calculation of intersection numbers, yielding effective degree bounds (Berczi, 2015, Berczi, 2010).

3.4 Nevanlinna Theory and Second Main Theorem

Complex analytic value-distribution methods are critical, especially in the "relative" setting:

• For families dominated by abelian torsors, truncated second main theorem estimates relate the growth of characteristics for entire curves to the counting function of divisor intersections, yielding sharp degeneracy properties (Brunebarbe, 2023).
• Branched covers and logarithmic settings invoke analogous Nevanlinna-theoretic and Ahlfors–Schwarz type arguments (Rousseau et al., 2015).

4. Principal Advances and Degree Bounds for Hypersurfaces

4.1 Classical Thresholds and Algebraic Differential Equations

For generic smooth projective hypersurfaces $X \subset \mathbb{P}^{n+1}$, the optimal threshold for the existence of global algebraic differential equations satisfied by every entire curve is $d \geq n+3$ (Merker, 2010), but pushing from differential equation existence to algebraic degeneracy and hyperbolicity requires stronger approaches.

4.2 Effective and Polynomial Bounds

Pioneering works established:

• Exponential bounds for degeneracy and hyperbolicity: $d \sim n^{8n}$ (Berczi, 2010), $d \sim n^{9n}$ (Berczi, 2015).
• Polynomial improvements through advanced Morse inequalities and slanted vector field techniques, culminating in bounds such as $d > \tfrac{153}{4} n^5$ for algebraic degeneracy and hyperbolicity for $d >\tfrac{153}{4} (2n-1)^5$ (Cadorel, 27 Jun 2024).
• Combinatorial approaches (Diverio–Merker–Rousseau, Merker–Ta) lowered the effective degree bounds to $d \geq (\sqrt{n} \log n)^n$ for degeneracy and $d \geq (n \log n)^n$ for hyperbolicity, with expectations—contingent on technical positivity conjectures—for further reductions to $d \sim 2^{5n}$ (Merker et al., 2019).

4.3 Logarithmic, Complement, and Family Analogs

Logarithmic analogues (pairs $(X, D)$) and the complements of generic hypersurfaces or their unions reach similar algebraic hyperbolicity results at sharp or near-sharp degrees $d = 2n$, modulo proper exceptional loci (Chen et al., 2022, Ascher et al., 1 Oct 2024). For families of maximal Albanese dimension, the relative GGL conjecture is established with Nevanlinna-theoretic estimates uniform in families (Brunebarbe, 2023).

5. Extensions: Hyperbolicity and Arithmetic Analogues

5.1 Jet-Semistability and Kobayashi Hyperbolicity

A directed variety (X, V) is algebraically jet-hyperbolic if all induced structures at every stage in the Semple tower are of general type modulo the projection. If so, full Kobayashi hyperbolicity follows: all entire curves are constant (Demailly, 2014, Demailly, 2015). Notably, very general hypersurfaces of degree $d \geq 2n+2$ in $\mathbb{P}^{n+1}$ are hyperbolic (Demailly, 2015).

5.2 Arithmetic Analogues

Analogues in arithmetic geometry suggest that analogs of GGL hold for rational points (Lang-Vojta conjectures), with similar value-distribution and height-bound phenomena appearing in families dominated by abelian torsors (Brunebarbe, 2023).

5.3 Hyperbolicity via Large Local Systems

The existence of a large complex local system—i.e., one whose pullback to every positive-dimensional subvariety remains non-isotrivial—forces the strong GGL conclusion, unifying entire curves, images of abelian varieties, and non-general-type subvarieties as the same Zariski-closed locus (Brunebarbe, 2022).

6. Open Problems, Recent Progress, and Future Directions

• Base Locus Problem: Achieving surjectivity to $X$ for the common zero locus of all higher-order jet differential operators remains the main obstacle for a full proof of GGL in arbitrary general type settings (Demailly, 2010, Cadorel, 27 Jun 2024).
• Sharp Degree Thresholds: Bridging the gap between known polynomial/combinatorial bounds and the conjectural $d \sim 2n$ “celestial horizon” is a persistent focus (Merker et al., 2019).
• Broader Geometric Classes: Generalizing to complete intersections, orbifolds, log pairs, and higher-dimensional bases for families.
• Arithmetic and Non-Archimedean Analogues: Developing analogues for rational and integral points, and using non-Archimedean methods (e.g., harmonic maps to Berkovich spaces).
• Higher Dimensional Exceptional Loci: Explicitly characterizing the exceptional locus in higher dimensions and the relation to linear/flex/bitangent-type rigidity phenomena (Chen et al., 2022).
• Positivity Conjecture on Thom Polynomials: Verification for all dimensions would yield polynomial degree bounds for algebraic degeneracy (Berczi, 2010).

7. Summary Table: Progress on Degree Bounds for Generic Hypersurfaces

Bound Type	Minimum Known Degree	Reference
Jet differentials exist	$d \geq n+3$ (optimal for existence)	(Merker, 2010)
Algebraic degeneracy	$d \geq \frac{153}{4} n^5$	(Cadorel, 27 Jun 2024)
Algebraic degeneracy	$d \geq (\sqrt{n} \log n)^n$	(Merker et al., 2019)
Hyperbolicity	$d \geq \frac{153}{4}(2n-1)^5$	(Cadorel, 27 Jun 2024)
Hyperbolicity	$d \geq (n \log n)^n$	(Merker et al., 2019)
(Conjectural, via positivity)	$d \geq 2^{5n}$	(Merker et al., 2019)

• For classical statements and formalism: (Demailly, 2010, Demailly, 2014, Berczi, 2010)
• On relative and family versions: (Brunebarbe, 2023, Rousseau et al., 2015)
• Polynomial and subexponential bounds for hypersurfaces: (Cadorel, 27 Jun 2024, Merker et al., 2019, Berczi, 2010, Berczi, 2015)
• On log and complement cases: (Chen et al., 2022, Ascher et al., 1 Oct 2024)
• On jet-semistability, strong general type, and hyperbolicity: (Demailly, 2015, Bei et al., 2022, Brunebarbe, 2022)

For further technical details, specific formulas, and analytic techniques, the referenced articles provide complete expositions, explicit formulas, and comprehensive proofs in the context of the latest progress on the (generalized) Green-Griffiths-Lang conjecture.