The Minimal Model Program for Enriques Pairs and a New Definition of Singular Enriques Varieties

Core Concepts

This mathematics research paper introduces the concept of "primitive Enriques varieties" as a new definition for singular Enriques manifolds and proves that the Minimal Model Program (MMP) holds for a log canonical pair of an Enriques manifold and an effective R-divisor.

Abstract

• Bibliographic Information: Denisi, F. A., Ríos Ortiz, Á. D., Tsanikas, N., & Xie, Z. (2024). MMP for Enriques pairs and singular Enriques varieties. arXiv preprint arXiv:2409.12054v2.
• Research Objective: This paper aims to investigate the application of the Minimal Model Program (MMP) to Enriques manifolds and characterize the resulting minimal models.
• Methodology: The authors introduce the concept of "primitive Enriques varieties" as singular analogs of Enriques manifolds. They utilize techniques from birational geometry, including quasi-étale covers, Galois actions, and the equivariant MMP, to lift the MMP from Enriques manifolds to their singular counterparts.
• Key Findings: The authors prove that any (KY + BY)-MMP for a log canonical pair (Y, BY), where Y is an Enriques manifold and BY is an effective R-divisor, terminates with a minimal model (Y′, BY′). This minimal model (Y′, BY′) is characterized by Y′ being a Q-factorial primitive Enriques variety with canonical singularities and BY′ being a nef and effective R-divisor on Y′.
• Main Conclusions: This paper establishes the validity of the MMP for Enriques manifolds, confirming the termination of flips conjecture for this class of varieties. The introduction and characterization of primitive Enriques varieties provide a new framework for studying singular Enriques varieties.
• Significance: This research significantly contributes to the field of algebraic geometry, particularly the study of Enriques varieties and the MMP. It opens up new avenues for investigating the birational geometry of these varieties and their singularities.
• Limitations and Future Research: The paper primarily focuses on the MMP for Enriques manifolds. Further research could explore the MMP for more general classes of varieties with numerically trivial canonical divisors or investigate the properties and classification of primitive Enriques varieties in greater detail.

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MMP for Enriques pairs and singular Enriques varieties

by Fran... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2409.12054.pdf

Deeper Inquiries

How does the concept of primitive Enriques varieties contribute to the broader understanding of singular varieties beyond the scope of Enriques manifolds?

Answer:
The concept of primitive Enriques varieties significantly advances our understanding of singular varieties in several ways:

Extending the IHS-Enriques Dichotomy: Just as Enriques surfaces are closely related to K3 surfaces, Enriques manifolds are understood as quotients of irreducible holomorphic symplectic (IHS) manifolds. Primitive Enriques varieties extend this relationship to the singular setting, providing a natural counterpart to primitive symplectic varieties (the singular analogue of IHS manifolds). This parallel structure allows for a more unified approach to studying both classes and exploring their interconnectedness.

MMP Stability and Termination of Flips: A key result of the paper is that primitive Enriques varieties, like their smooth counterparts, are stable under the operations of the Minimal Model Program (MMP). This stability is crucial for studying the birational geometry of these varieties. Moreover, the authors prove the termination of flips conjecture for Enriques manifolds, a significant result as this conjecture remains open in higher dimensions. This suggests that the special structure of Enriques varieties, even in the singular case, imposes strong constraints on their birational behavior.

New Insights into Singularities: By studying the singularities that can arise on primitive Enriques varieties, the authors shed light on a broader class of singularities on varieties with numerically trivial canonical class. This is particularly relevant as the singularities allowed on primitive Enriques varieties are not necessarily quotients of singularities appearing on IHS manifolds, highlighting the richness and complexity of this new class.

Foundation for Further Research: The introduction of primitive Enriques varieties opens up several avenues for future research. For example, it would be interesting to investigate their moduli spaces, deformation theory, and potential connections to other types of singular varieties.
In essence, primitive Enriques varieties provide a new lens through which to study singular varieties with connections to symplectic geometry. Their properties and behavior under the MMP offer valuable insights into the broader landscape of algebraic varieties.

Could there be alternative definitions of singular Enriques varieties that might lead to different outcomes or insights regarding the MMP?

Answer:
Yes, alternative definitions of singular Enriques varieties could certainly lead to different outcomes and insights regarding the MMP. The paper focuses on "primitive" Enriques varieties, defined as quasi-étale quotients of primitive symplectic varieties by non-symplectic group actions. This definition naturally extends the relationship between smooth Enriques and IHS manifolds to the singular setting. However, other approaches are possible, each with potential advantages and drawbacks:

Weakening the Quotient Condition: One could consider varieties with a finite (not necessarily quasi-étale) cover by a primitive symplectic variety, allowing for more general singularities. This broader definition might encompass a wider range of varieties but could make it more challenging to control the behavior of the MMP, particularly regarding the singularities of the resulting models.

Focusing on Numerical Invariants: An alternative approach could involve defining singular Enriques varieties based on numerical invariants, such as having numerically trivial canonical class and specific Hodge numbers. This approach might be more flexible but could lead to a less geometrically intuitive definition and might not guarantee good behavior under the MMP.

Logarithmic Enriques Varieties: As mentioned in the paper, an alternative definition of singular Enriques varieties, called "logarithmic Enriques varieties," has been introduced independently. These are defined as varieties with a special type of covering by a log symplectic variety. Exploring the interplay between these two definitions and their respective implications for the MMP would be an interesting research direction.
The choice of definition ultimately depends on the specific goals and questions being considered. The "primitive" definition used in the paper is well-suited for studying the MMP and extending results from the smooth case. However, exploring alternative definitions could reveal new aspects of the geometry of these varieties and their relationship to other classes of singular varieties.

What are the implications of this research for the classification and understanding of higher-dimensional algebraic varieties with similar geometric properties?

Answer:
This research has significant implications for the classification and understanding of higher-dimensional algebraic varieties, particularly those with numerically trivial canonical class:

Expanding the Classification Framework: The introduction of primitive Enriques varieties adds a new building block to the classification program for higher-dimensional varieties. By understanding their properties and behavior under the MMP, we gain a better grasp of the broader landscape of varieties with trivial canonical class, which are central to the Minimal Model Program.

New Examples and Constructions: The paper provides various examples of primitive Enriques varieties, enriching the known families of varieties with these properties. These examples can serve as test cases for conjectures and inspire the development of new construction techniques for varieties with similar geometric features.

Insights from Symplectic Geometry: The close relationship between primitive Enriques varieties and symplectic varieties allows us to leverage tools and techniques from symplectic geometry to study their structure and properties. This cross-fertilization of ideas can lead to new insights and a deeper understanding of both fields.

Asymptotic Invariants and Positivity: The study of asymptotic invariants, such as the ample and movable cones of divisors, is crucial for understanding the birational geometry of algebraic varieties. The authors' results on the duality of cones for Enriques manifolds provide valuable information about the positivity properties of divisors on these varieties and can potentially be extended to other classes with similar geometric features.

Motivating Further Exploration: This research raises intriguing questions and opens up new avenues for investigation. For instance, exploring the moduli spaces of primitive Enriques varieties, their relationship to other classes of varieties, and the development of analogous theories for varieties with different types of coverings are all promising directions for future research.
Overall, this work provides a significant step forward in our understanding of higher-dimensional algebraic varieties. By introducing and studying primitive Enriques varieties, the authors not only contribute to the classification program but also provide new tools and perspectives for investigating the rich and complex world of algebraic varieties with numerically trivial canonical class.