Big Picard Type Theorem Overview

• Big Picard Type Theorem is a generalization that extends the classical removability of isolated singularities to higher-dimensional, moduli, and non-archimedean contexts.
• It utilizes advanced methodologies such as logarithmic Higgs bundles, Finsler metrics, and Nevanlinna theory to achieve holomorphic extensions and establish hyperbolicity properties.
• The theorem provides critical insights into moduli stacks, jet differential techniques, and arithmetic hyperbolicity, influencing both complex analysis and algebraic geometry.

The Big Picard Type Theorem is a powerful generalization of the classical big Picard theorem from complex analysis, extending the paradigm of removability of isolated singularities for holomorphic maps to higher-dimensional and arithmetic-geometric contexts. This concept features prominently in the paper of moduli spaces, transcendental value distribution theory, and complex and non-archimedean geometry, driving advances in both the structure theory of moduli and questions of arithmetic and algebraic hyperbolicity.

1. Classical Big Picard Theorem and General Framework

The classical Big Picard theorem addresses the extension of holomorphic maps $f:\Delta^*\to \mathbb{P}^1$ (where $\Delta^*=\{t\in\mathbb{C}:0<|t|<1\}$ is the punctured unit disk) omitting three points in $\mathbb{P}^1$. The theorem asserts that such $f$ extends holomorphically across the puncture, i.e., $f:\Delta\to\mathbb{P}^1$ exists, with $\Delta$ the full disk. This can be recast in modern terms: if $U\subset\mathbb{P}^1$ is hyperbolically embedded (omitting at least three points), any map from $\Delta^*$ to $U$ extends across the puncture.

Higher-dimensional analogues generally seek removability criteria for holomorphic curves or more general analytic morphisms into quasi-projective (or more general analytic) targets that are “hyperbolically embedded” or satisfy a suitable analog of value-omission or curvature negativity.

2. Big Picard Theorem for Moduli Spaces of Polarized Manifolds

A central advancement is the extension of the Big Picard property to moduli spaces of polarized algebraic manifolds, particularly those parameterizing smooth projective varieties with semi-ample canonical sheaf (Deng, 2019). Here, the setting is a smooth projective family

$f_U:U\to V,\qquad L\to U$

with $L$ relatively ample and $K_{U/V}$ semi-ample, over a quasi-projective base $V$. The moduli map $\varphi_U: V\to P_h$ (to the coarse moduli space $P_h$ for fixed Hilbert polynomial $h$) is assumed quasi-finite.

The key result is:

• For any projective compactification $Y\supset V$ (with $D=Y\setminus V$ a simple normal crossings divisor), every holomorphic $f:\Delta^*\to V$ extends to $\tilde{f}:\Delta\to Y$.

The proof employs the construction of a logarithmic Higgs bundle $(E,\theta)$ and a big line bundle $L$ on $Y$, combined with Finsler pseudometrics arising from the iteration of the Kodaira–Spencer morphism. The crucial step is to show that these Finsler metrics satisfy a negative curvature estimate, and then apply a Nevanlinna-theoretic extension criterion: if $-dd^c\log |f'(t)|_h^2\geq f^*\omega$ for $h$ a Finsler metric and some Kähler form $\omega$ on $Y$, then $f$ extends holomorphically.

This result immediately implies:

• Brody hyperbolicity (no nonconstant entire holomorphic curves in $V$)
• Borel hyperbolicity (every holomorphic map from a quasi-projective variety to $V$ is algebraic).

Moreover, the result generalizes to families of log Calabi–Yau pairs, under suitable injectivity of the Kodaira–Spencer map.

3. Big Picard Phenomena in Moduli Stacks and Strata

The stratified structure of moduli stacks and their implications for hyperbolicity and extension properties have been elucidated in the context of stacks of stable minimal models (Shentu, 24 Jun 2025). For each stratum $S$ in a birationally admissible stratification of the Deligne–Mumford stack $M_{\rm slc}(d,\Phi_c,\Gamma,\sigma)$, the closure $\overline{S}$ with boundary $\partial S$ forms a Picard pair:

• For any holomorphic $\gamma:\Delta^*\to S^{\rm an}$, up to reparametrization $z\mapsto z^n$, either $\gamma$ extends to $\Delta\to \overline{S}^{\rm an}$, or $\gamma$ accumulates on the boundary.

Over each stratum, the existence of a simultaneous log-birational model for the universal family allows for the construction of suitable Higgs bundles and Finsler metrics, leading to curvature and extension estimates analogous to those above. Consequently, each stratum is shown to be Borel hyperbolic and Brody hyperbolic, displaying strong analytic-arithmetic rigidity and ruling out punctured disk or entire maps with unremovable singularities.

4. Extensions to Non-Archimedean and Berkovich Analytic Geometry

The Big Picard property has analogues in non-archimedean analytic geometry—both in Berkovich and rigid analytic settings (Okuyama, 2019, Huynh et al., 2020). For Berkovich projective space $\mathsf{P}^N$ over a complete non-archimedean field $K$, if an open subset $\Omega\subset \mathsf{P}^N$ has the property that the family $\mathrm{Mor}(D^*,\Omega)$ is Montel-normal—i.e., every sequence has a subsequence converging locally uniformly on affinoids—then every morphism $f:D^*\to\Omega$ extends to $D\to \mathsf{P}^N$. This formalizes the absence of isolated essential singularities under a Montel-style normality condition.

In the context of jet differentials, the Big Picard theorem further states: for $X$ a smooth projective variety over a non-archimedean field and $D$ a snc divisor, any rigid analytic map $f:\A(0,R]\to X\setminus D$ with nontrivial pullback under a global logarithmic jet-differential extends to the closed disk (Huynh et al., 2020). Jet theory, via the Green–Griffiths bundle and Nevanlinna-type estimates, supplies the fine analytic control necessary to derive extension criteria.

These principles subsume prior results by Cherry and Ru, permit proofs of non-archimedean Ax–Lindemann theorems, and establish analytic Borel hyperbolicity for general-type subvarieties of abelian varieties.

5. Metric and Hyperbolicity Criteria: Methods and Implications

A frequent underpinning of Big Picard type results is the existence of negatively curved Finsler pseudometrics on log-tangent bundles. This metric approach centers on the following scheme:

• Construction of a continuous Finsler pseudometric $h$ on $T_X(-\log D)$ such that for any holomorphic curve $f:\Delta^*\to X\setminus D$, the Ricci form of the pullback metric satisfies:

$$\mathrm{Ric}(f^*h) = -dd^c\log|f'(z)|^2_{f^*h} \leq -\lambda f^* \omega$$

for $\lambda>0$ and a Kähler form $\omega$.

The combination of strong curvature negativity and the Ahlfors–Schwarz lemma or Nevanlinna-theoretic argument then yields removability of singularities for such $f$, i.e., the Big Picard property. This methodology underpins results within both complex and non-archimedean settings (Deng et al., 2019, Deng, 2019, Huynh et al., 2020).

The construction of such metrics is achieved through refined Viehweg–Zuo Higgs bundle techniques, including the iteration of Kodaira–Spencer maps, the use of ample line bundles, and comparison with Hodge metrics. The presence of such Finsler metrics is itself evidence for algebraic and geometric hyperbolicity properties (Brody, Borel, algebraic hyperbolicity).

6. Generalizations, Corollaries, and Arithmetic Significance

Big Picard type theorems have a spectrum of corollaries:

• Borel hyperbolicity: any holomorphic map from an algebraic variety (or stack) to the target is algebraic; holomorphic families arise algebraically.
• Brody hyperbolicity: nonexistence of nontrivial entire maps from $\mathbb{C}^1$.
• Algebraic hyperbolicity: Lower bounds on intersection numbers and genus for curves not contained in the boundary, generalizing Arakelov-type inequalities.
• Diophantine finiteness: A plausible implication is that, when combined with arithmetic methods, the Big Picard property can imply finiteness of integral points and rigidity of maps from arithmetic varieties.

Furthermore, the results generalize classical and higher-dimensional Picard theorems, often recovering the extension property for maps omitting sufficiently ample or numerous divisors in projective space as special cases.

7. Kobayashi Hyperbolicity, Essential Singularities, and Value Distribution

Underlying much of the theory is the relationship to Kobayashi (and Brody) hyperbolicity (Okuyama, 2014). For a holomorphic curve $f:D^*\to Y$ into a hyperbolically embedded subspace $Y\subset Z$, the Big Picard property manifests as the nonexistence of isolated essential singularities—every such $f$ with non-escaping image extends across $0$.

These results leverage the Kobayashi pseudometric and generalizations of the Lehto–Virtanen theorem, as well as rescaling principles enabling the extraction of entire or punctured entire curves from sequences approaching essential singularities. In higher dimensions and singular targets, the analytic approach via pseudometrics is crucial, supplementing classical Nevanlinna theory.

This synthesis of analytic, algebro-geometric, and arithmetic methods defines the modern scope of Big Picard type theorems, establishing them as central tools in moduli theory, hyperbolic geometry, and transcendence.