Core Concepts

This mathematics research paper explores the connection between the Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces and the existence of Shokurov polytopes, proposing a new conjecture that might be more approachable for proving the cone conjecture.

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arxiv.org

Li, Z., & Zhao, H. (2024). On the relative Morrison-Kawamata cone conjecture. arXiv preprint arXiv:2206.13701v5.

This paper investigates the relative Morrison-Kawamata cone conjecture, which predicts the structure of movable and ample cones of Calabi-Yau fiber spaces, aiming to relate it to the existence of Shokurov polytopes as a potentially more tractable approach.

Key Insights Distilled From

by Zhan Li, Han... at **arxiv.org** 10-24-2024

Deeper Inquiries

Yes, the techniques introduced in the paper, particularly the interplay between Shokurov polytopes and the Morrison-Kawamata cone conjecture, hold promise for applications beyond the immediate scope of the cone conjecture. Here's why:
Understanding Birational Contractions: The core of the paper lies in analyzing how weak log canonical models behave under perturbations of the divisor within a Shokurov polytope. This analysis provides insights into the geometry of birational contractions associated with these models. Such understanding could be valuable in studying other problems where birational contractions play a central role, such as:
Finiteness of Minimal Models: While the cone conjecture itself has implications for the finiteness of minimal models, the techniques used to relate it to Shokurov polytopes could potentially lead to alternative approaches or refinements of existing results.
Structure of the Minimal Model Program: The decomposition of the cone into chambers corresponding to different minimal models, as suggested by the cone conjecture, hints at a deeper structure underlying the Minimal Model Program. The methods employed in the paper could offer new perspectives on this structure.
Exploring Cones Beyond Ample and Movable: The paper focuses on the ample cone and the movable cone. However, the techniques could potentially be adapted to study other cones relevant to birational geometry, such as the nef cone or the pseudo-effective cone. This could lead to new conjectures and results about the structure of these cones under birational transformations.
Generalizations to Singularities: The paper primarily deals with klt singularities. Investigating whether similar relationships between Shokurov polytopes and cone-like structures exist for more general singularities, such as lc singularities, could be a fruitful direction for future research.

It's certainly plausible that counterexamples to the cone conjecture might exist when the existence of good minimal models is not guaranteed. Here's why:
Good Minimal Models and Finiteness: A key motivation behind the cone conjecture is the expectation that the finiteness of minimal models is closely tied to the existence of rational polyhedral fundamental domains. Good minimal models, with their semi-ampleness property, naturally lend themselves to constructing such finite domains.
Lack of Finiteness: In situations where good minimal models are not known to exist, there's a possibility of encountering an "accumulation" of birational models that could obstruct the existence of a finite fundamental domain. This potential lack of finiteness might manifest as:
Infinitely Many Minimal Models: Without the semi-ampleness provided by good minimal models, there might be infinitely many birational models that are "minimal" in a weaker sense, leading to an infinite fundamental domain.
Non-Polyhedral Fundamental Domain: Even if the number of minimal models is finite, the fundamental domain might no longer be describable by a finite set of inequalities, resulting in a non-polyhedral shape.
Exploring the Boundary Cases: Investigating potential counterexamples in cases where the existence of good minimal models is not guaranteed, such as for varieties with more general singularities or in higher dimensions, could provide valuable insights into the limits of the cone conjecture and the subtle interplay between minimal models and the structure of cones.

The concept of fundamental domains in the context of the cone conjecture is a beautiful illustration of how symmetry, represented by group actions, can impose a rigid structure on geometric objects. This theme resonates deeply across various areas of mathematics:
Geometry and Topology:
Hyperbolic Geometry: The action of discrete groups on the hyperbolic plane leads to the classic example of fundamental domains, such as the modular group and its action on the upper half-plane. These domains tessellate the hyperbolic plane and provide a powerful tool for studying hyperbolic surfaces.
Lie Groups and Symmetric Spaces: Fundamental domains for the action of discrete subgroups on Lie groups and symmetric spaces are central to understanding their geometry and topology. They play a crucial role in representation theory, number theory, and the study of automorphic forms.
Number Theory:
Lattices and Quadratic Forms: The action of the orthogonal group on the space of quadratic forms leads to the notion of fundamental domains for lattices. These domains are essential for classifying quadratic forms and studying their arithmetic properties.
Modular Forms: Modular forms, which are functions with remarkable symmetry properties, are defined on fundamental domains for the action of certain subgroups of the modular group. These domains provide a natural setting for studying the intricate relationship between modular forms and number theory.
Dynamical Systems:
Ergodic Theory: Fundamental domains arise in the study of dynamical systems with symmetries. They provide a way to understand the long-term behavior of orbits under the action of a group and are used to prove ergodic theorems.
In essence, the cone conjecture, by linking the existence of fundamental domains to the finiteness of minimal models, suggests a profound connection between the algebraic structure of varieties and the geometric structure of cones under the action of birational transformations. This connection exemplifies the broader principle that symmetry, often encoded by group actions, can reveal hidden order and provide powerful tools for understanding complex mathematical objects.

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