Torically Hyperbolic Varieties: Insights

• Torically hyperbolic varieties are complex algebraic varieties defined by combining Kobayashi and Demailly hyperbolicity with toric and tropical geometric structures.
• The framework employs essential projective hyperplane complements and angle set decompositions to achieve pair-of-pants models and homotopy equivalences.
• Robust combinatorial criteria and deformation theory ensure persistent hyperbolicity via genus–degree inequalities and controlled bad loci in toric settings.

A torically hyperbolic variety is a complex algebraic variety characterized by hyperbolicity properties defined in the context of toric and @@@@2@@@@. The theory links the hyperbolicity notions of Kobayashi, Demailly, and their adjoint-bundle incarnations with special combinatorial and topological structures arising from toric degenerations, angle spaces, and tropical geometry. The term encompasses multiple precise notions: Kobayashi hyperbolicity for toric complements, algebraic (or Demailly) hyperbolicity in the sense of genus–degree inequalities, and new topological models based on pair-of-pants decompositions in the toric and tropical setting. Recent advances provide both combinatorial and topological criteria for identifying and analyzing torically hyperbolic varieties, especially within the categories of projective toric varieties, their very affine and log-general type degenerations, and their open subvarieties.

1. Notions of Hyperbolicity in the Toric Context

Several hyperbolicity notions are relevant for torically hyperbolic varieties:

• Kobayashi Hyperbolicity: A complex manifold $X$ is Kobayashi hyperbolic if its Kobayashi pseudodistance is nondegenerate, i.e., $X$ admits no entire holomorphic maps $\mathbb{C}\to X$ (Moraga et al., 2024, Yoo et al., 2021).
• Kobayashi Hyperbolic Embedding: For an open subvariety $Y\subset X$, the inclusion $Y\hookrightarrow X$ is a Kobayashi hyperbolic embedding if $Y$ is hyperbolic and the induced pseudodistance does not collapse at the boundary (Yoo et al., 2021).
• Algebraic (Demailly) Hyperbolicity: A projective variety $X$ with an ample divisor $H$ is algebraically hyperbolic if there exists $\varepsilon>0$ such that every nonconstant morphism $f:C\to X$ from a smooth projective curve $C$ satisfies $2g(C)-2\geq \varepsilon \cdot \deg f^*H$ (Moraga et al., 2024). This condition rules out a wealth of rational and low-genus curves.
• Pseudo-Hyperbolicity: A linear system $|H|$ on $X$ is pseudo-hyperbolic modulo $Z$ if the genus–degree bound holds for all maps $f:C\to D\subset X$ not contained in the proper subset $Z\subset D$ (Moraga et al., 2024).

In toric settings, these notions interact in subtle ways, often mediated by the combinatorics of divisors and the geometry of toric degenerations.

2. Definitional Criteria and Local Models

The structural foundation of torically hyperbolic varieties leverages two key ingredients: essential projective hyperplane complements, and the geometry of angle sets (as “pair-of-pants” domains).

• Essential Projective Hyperplane Complements: Given $H_0,\dots,H_n$ linear forms on $\mathbb{P}^d$, the complement $Y=\mathbb{P}^d\setminus \bigcup_{j=0}^n\{H_j=0\}$ is essential if the induced map

$$\varphi:\mathbb{P}^d\setminus\bigcup_{j=0}^n\{H_j=0\}\to (\mathbb{C}^\times)^n$$

is an embedding whose image $Z\subset T$ is affine-linear (Elmaazouz et al., 20 Jan 2026).

• Angle Sets: For $Z\subset T=(\mathbb{C}^\times)^n$, the angle set is $\Theta=\mathrm{ang}(Z(\mathbb{C}))\subset (S^1)^n$, where $S^1$ is the unit circle. The angle map is a homotopy equivalence in these cases (Elmaazouz et al., 20 Jan 2026).

A semistable degeneration model $\mathcal{X}\to U$ is torically hyperbolic if its special fiber at $t=0$ is a normal crossing divisor, with all irreducible components (more precisely: their punctured opens) isomorphic to essential projective hyperplane complements.

Definition (Elmaazouz et al., 20 Jan 2026): A smooth variety $X$ is torically hyperbolic if it appears as a general fiber of a torically hyperbolic model—a flat semistable degeneration whose irreducible special fiber strata are essential projective hyperplane complements.

3. Combinatorial Characterization and Deformation Theory

A core insight is that toric (or toroidal) hyperbolicity is largely governed by explicit combinatorial criteria. This yields both Zariski open and closed structural results:

• Kobayashi Hyperbolic Embedding Criterion: For a projective toric variety $X$ and a Cartier divisor $D$ with full-dimensional Newton polytope $P_D$, there exists a Zariski closed subset $\mathcal{Y}\subset|D|$ such that for $D_0\in |D|\setminus \mathcal{Y}$, the inclusion $T_N\setminus D_0\hookrightarrow X$ is a Kobayashi hyperbolic embedding. The “bad locus” $\mathcal{Y}$ is constructed via combinatorial slice data on the support lattice $S=P_D\cap M$ (Yoo et al., 2021).
• Persistence under Deformation: If $D_0$ is torically hyperbolic, then for any algebraic family $\{D_t\}$ parametrized by a curve avoiding $\mathcal{Y}$ at finitely many points, hyperbolicity persists for all $t$ outside a finite set (Yoo et al., 2021).

This Zariski-open phenomenon is verified in classical contexts (e.g., for $$\mathbb{P}^n\setminus\{\text{degree }n\text{ hypersurface}\}$$), as well as for multidegree and bidegree hypersurfaces in products and Hirzebruch surfaces. The explicit combinatorics required to check the bad locus are detailed for low-dimensional examples, with open questions remaining for higher-dimensional or more degenerate toric varieties.

4. Pair-of-Pants Decompositions and Topological Models

A notable development is the topological decomposition of torically hyperbolic varieties into angular pair-of-pants models, extending classical constructions from Riemann surfaces to higher dimensions.

• Local Building Block (Theorem A): Every affine-linear very affine subvariety $Z\subset (\mathbb{C}^\times)^n$ is homotopy equivalent to its angle set $\Theta\subset (S^1)^n$. The angle map $\mathrm{ang}: Z(\mathbb{C})\to \Theta$ is a homotopy equivalence (Elmaazouz et al., 20 Jan 2026).
• Kummer Coverings: The homotopy equivalence persists under finite toric coverings (Kummer coverings), ensuring that the local angular models are robust under toric base change (Elmaazouz et al., 20 Jan 2026).
• Global Gluing via Kato–Nakayama and Tropical Degenerations: In a semistable degeneration $\mathcal{X}\to U$ of a torically hyperbolic variety, each stratum is covered by its Kato–Nakayama space, and their angle sets $\Theta_\sigma^{\log}$ glue along the dual complex $\Sigma$. The resulting homotopy colimit is homotopy equivalent to a general fiber $X_t$, yielding a canonical “pair-of-pants” decomposition of $X$ via its toric degeneration data (Elmaazouz et al., 20 Jan 2026).

When $X$ is a generic complete intersection in $(\mathbb{C}^\times)^n$, there exists a degeneration so that the tropicalization is a stable intersection of smooth hypersurfaces; the dual complex $\Sigma$ parametrizes the gluing of local angle data, and the fundamental group and higher homotopy structure of $X$ is recovered combinatorially.

5. Algebraic Hyperbolicity and Genus–Degree Inequalities

In the projective toric setting, algebraic hyperbolicity is often verified via combinatorial and intersection-theoretic genus–degree bounds.

• Haase–Ilten Criterion: For a surface $S$ in a smooth projective toric threefold $X$, if $(D,E)$ are basepoint-free torus-invariant divisors with $D$ big and “connected sections,” then for $C\subset S$ not contained in the toric boundary,

$2g(C) - 2 \geq C\cdot (E + K_X)$

and the genus of each boundary curve $S\cap D_\rho$ equals the number of interior lattice points of the associated polytope face (Robins, 2021, Haase et al., 2019).

• Hyperbolicity of Very General Surfaces: For explicit toric threefolds (e.g., $\mathbb{P}^2\times\mathbb{P}^1$, $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$), the very general surface in an ample linear system is algebraically hyperbolic above explicit degree thresholds (Coskun et al., 2019).
• Adjoint Bundle Conjecture: For a smooth projective toric variety $X$ of dimension $n$ and ample $L$, a very general element of $|K_X + (3n+1)L|$ is algebraically hyperbolic. For Gorenstein toric threefolds, $|K_X + 9L|$ is hyperbolic (Moraga et al., 2024).

A plausible implication is that combinatorial data (fan, polytope, section graphs) together with connected sections property suffices to classify nearly all algebraically hyperbolic surfaces in explicit toric settings.

6. Examples, Applications, and Open Problems

• Concrete Examples:
  • In $\mathbb{P}^2$, the complement of a very general conic is torically hyperbolic (Yoo et al., 2021).
  • In $\mathbb{P}^3$, the complement of a cubic satisfies a higher codimension criterion for toric hyperbolicity (Yoo et al., 2021).
  • For surfaces in $\mathbb{P}^2\times\mathbb{P}^1$ or $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of bi- or tridegree $(a,b)$ with $a,b$ large, algebraic hyperbolicity holds generically (Coskun et al., 2019).
• Tropical Complete Intersections: Tropicalization techniques ensure that a generic complete intersection with smooth tropicalization admits a torically hyperbolic model and topological decomposition by angle sets (Elmaazouz et al., 20 Jan 2026).
• Deformation Rigidity: The combinatorial criteria for the “bad locus” ensure that toric hyperbolicity is stable under algebraic deformations modulo a finite exceptional set (Yoo et al., 2021).
• Open Questions:
  • An explicit combinatorial description of the bad locus for higher-dimensional toric varieties remains open.
  • The relation between the algebraic hyperbolicity of toric complements and broader conjectures (e.g., Green–Griffiths–Lang) is under investigation, especially regarding the precise threshold for general type behavior.
  • Extension of these techniques to varieties with higher Picard rank or to toric fourfolds is a prospective direction (Robins, 2021).

7. Schematic Relationships Among Torically Hyperbolic Varieties

Notion	Characterization	Main Results / Techniques
Kobayashi hyperbolicity	No entire curves	Combinatorial “slice” conditions, Zariski bad locus (Yoo et al., 2021)
Kobayashi hyperbolic embedding	Hyperbolicity + boundary separation	Explicit in complete linear systems
Algebraic (Demailly) hyperbolicity	$2g-2\geq \varepsilon\deg_H(C)$	Section–dominating systems, genus–degree inequalities (Haase et al., 2019, Coskun et al., 2019)
Topological pair-of-pants	Homotopy equivalence to angle set	Angle map, Kato–Nakayama, tropicalization (Elmaazouz et al., 20 Jan 2026)
Adjoint hyperbolicity	Thresholds in $|K_X + (3n+1)L|$	Induction in dimension, syzygy bundles, vanishing (Moraga et al., 2024)

This synthesis demonstrates that the study of torically hyperbolic varieties is governed by an interplay of combinatorial, topological, and intersection-theoretic data. The ultimate objective is to classify and analyze hyperbolicity in the presence of toric symmetry, tropical degenerations, and their implications for both the geometry and topology of algebraic varieties. The rapidly evolving framework continues to connect finer moduli-theoretic, topological, and combinatorial inputs in new classes of varieties and degenerations.