Core Concepts

Lang's conjecture, suggesting that rational curves and Abelian varieties are the sole obstructions to hyperbolicity in projective manifolds, is shown to be a consequence of the abundance conjecture and the existence of sufficiently many rational curves on specific varieties.

Abstract

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arxiv.org

Broder, K., & Campana, F. (2024). Remarks on Lang’s Conjectural Characterisation of Hyperbolicity of Projective Manifolds. arXiv, 2410.06402v1.

This research paper explores the validity of Lang's conjecture, which posits that the only obstructions to a projective manifold being hyperbolic are the presence of rational curves or images of Abelian varieties. The authors aim to demonstrate that this conjecture can be derived from the abundance conjecture and the assumption of ample rational curves on certain types of varieties.

Deeper Inquiries

This paper cleverly weaves together concepts of hyperbolicity, minimal model program, and entire curves to investigate the geometry of algebraic varieties. The techniques employed offer potential avenues for exploring other geometric properties beyond hyperbolicity. Here are a few possibilities:
Structure of Special Varieties: The paper heavily relies on the interplay between the existence of rational curves, the abundance conjecture, and the classification of special varieties. These ideas could be further employed to study the structure of weakly special varieties more deeply. For instance, one might investigate the relationship between the geometry of the fibers of the Moishezon-Iitaka fibration and the presence of entire curves with specific properties.
Kobayashi Pseudodistance and Related Metrics: The paper establishes connections between the Kobayashi pseudodistance, the existence of rational curves, and the structure of the exceptional locus. This suggests further exploration of how properties of the Kobayashi pseudodistance (or related metrics like the Bergman metric) can be used to characterize geometric features of algebraic varieties. For example, one could investigate whether the asymptotic behavior of the Kobayashi metric near special subvarieties reveals information about their geometry.
Generalizations to Higher-Dimensional Varieties: While the paper focuses on projective manifolds, many of the techniques and concepts used have natural generalizations to higher-dimensional varieties and even singular spaces. Extending the results and techniques to these more general settings could lead to new insights into the geometry of these spaces. For instance, one might investigate the relationship between the singularities of a variety and the structure of its exceptional locus.
Connections to Other Notions of Hyperbolicity: Beyond Brody hyperbolicity, various other notions of hyperbolicity exist in complex geometry, such as Kobayashi hyperbolicity and algebraic hyperbolicity. Investigating the connections between these different notions, particularly in light of the results presented in the paper, could be fruitful. For example, one might explore whether the presence of certain types of special subvarieties obstructs other forms of hyperbolicity.
By adapting and extending the techniques presented in this paper, researchers can gain a deeper understanding of the intricate relationship between algebraic and geometric properties of algebraic varieties.

While the abundance conjecture and the existence of rational curves provide a powerful framework for understanding obstructions to hyperbolicity, exploring alternative characterizations that rely on different geometric or topological invariants is an intriguing question. Here are a few potential directions:
Holomorphic Sectional Curvature: As discussed in the paper, there is a strong connection between negative holomorphic sectional curvature and hyperbolicity. Investigating whether weaker curvature conditions (e.g., non-positive holomorphic sectional curvature outside a subvariety) could still impose restrictions on the presence of entire curves might provide alternative characterizations.
Nevanlinna Theory and Value Distribution: Nevanlinna theory offers tools to study the value distribution of holomorphic maps. It might be possible to formulate obstructions to hyperbolicity based on the growth and distribution of entire curves within a projective manifold, potentially circumventing the need for the abundance conjecture or the existence of rational curves.
Deformations and Moduli Spaces: Studying how hyperbolicity behaves under deformations of the complex structure of a projective manifold could lead to new insights. For instance, one might investigate whether the presence of special subvarieties obstructing hyperbolicity is stable under deformations or if it imposes restrictions on the geometry of the moduli space.
Cohomological Obstructions: Exploring whether cohomological invariants, such as Hodge numbers or Chern classes, could detect the presence of obstructions to hyperbolicity is another possibility. For example, certain inequalities involving these invariants might be indicative of the existence of entire curves.
Dynamical Characterizations: Instead of focusing solely on the existence of entire curves, one could investigate the dynamics of holomorphic self-maps on projective manifolds. The presence of special subvarieties might be reflected in the dynamical behavior of these maps, potentially leading to alternative characterizations of obstructions to hyperbolicity.
Developing alternative characterizations of obstructions to hyperbolicity would provide a more comprehensive understanding of this fundamental concept and its relationship to other aspects of algebraic geometry.

The findings in this paper, particularly the construction of examples with prescribed exceptional loci, have interesting implications for the study of complex dynamics on projective manifolds:
Understanding the Fatou and Julia Sets: In complex dynamics, the Fatou set of a holomorphic map is where the dynamics are stable, while the Julia set is where chaotic behavior occurs. The exceptional locus, as characterized in the paper, can be thought of as a dynamically "special" set. The paper's findings suggest that the structure of the exceptional locus can influence the geometry and topology of the Fatou and Julia sets of holomorphic maps on projective manifolds.
Construction of New Dynamical Examples: The construction techniques used in the paper, particularly the use of blow-ups and finite covers, could be adapted to construct new examples of projective manifolds with interesting dynamical behavior. For instance, one could potentially construct examples where the Julia set is contained in a prescribed subvariety or exhibits specific topological properties.
Connections to Arithmetic Dynamics: The interplay between hyperbolicity and arithmetic properties of algebraic varieties is a rich area of research. The findings in this paper, particularly the connection between special subvarieties and entire curves, could have implications for understanding the distribution of rational points on algebraic varieties and the dynamics of iterated maps over number fields.
Exploring the Dynamics on Weakly Special Varieties: The paper highlights the importance of weakly special varieties in the context of hyperbolicity. Investigating the dynamics of holomorphic maps specifically on weakly special varieties could reveal new phenomena and connections between the dynamical and geometric properties of these spaces.
Generalizing Dynamical Results to Singular Spaces: While the paper focuses on smooth projective manifolds, the techniques and concepts used could potentially be extended to study the dynamics of holomorphic maps on singular complex spaces. This would open up new avenues for research and could lead to a deeper understanding of the interplay between singularities and complex dynamics.
By shedding light on the relationship between special subvarieties and hyperbolicity, this paper provides valuable tools and insights for researchers studying complex dynamics on projective manifolds and beyond.

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