Compactifying Quasi-Projective Lagrangian Fibrations Using Holomorphic Symplectic Varieties: Criteria and Applications

Core Concepts

This paper presents a framework for compactifying quasi-projective Lagrangian fibrations using holomorphic symplectic varieties, providing criteria for compactification and exploring applications to fibrations of geometric origin.

Abstract

• Bibliographic Information: Saccà, G. (2024). Compactifying Lagrangian fibrations. arXiv preprint arXiv:2411.06505v1.
• Research Objective: To establish a general framework for compactifying quasi-projective Lagrangian fibrations with holomorphic symplectic varieties, examining the conditions under which such compactifications exist and are smooth.
• Methodology: The paper utilizes techniques from algebraic geometry, particularly the minimal model program (MMP) with scaling, to construct compactifications. It leverages the properties of holomorphic symplectic forms, Hodge theory, and the geometry of Lagrangian fibrations.
• Key Findings:
  • The paper proves a theorem providing sufficient conditions for the existence of a Q-factorial terminal symplectic compactification of a quasi-projective Lagrangian fibration.
  • It introduces the concept of "extendable holomorphic forms" and demonstrates their significance in compactification criteria.
  • The study establishes the existence of a relative Albanese fibration associated with a Lagrangian fibration admitting local sections, extending previous results.
  • It applies the framework to various geometric examples, including intermediate Jacobian fibrations and torsors over relative Albanese varieties.
• Main Conclusions: The paper successfully establishes a theoretical framework for compactifying a class of Lagrangian fibrations. It demonstrates the existence of such compactifications under specific conditions and explores their properties. The research offers valuable insights into the geometry and topology of Lagrangian fibrations and their applications in algebraic geometry.
• Significance: This work contributes significantly to the study of Lagrangian fibrations, a crucial topic in algebraic geometry with connections to mirror symmetry and integrable systems. The framework and results presented have implications for understanding the moduli spaces of various geometric objects and constructing new examples of holomorphic symplectic varieties.
• Limitations and Future Research: The paper primarily focuses on Lagrangian fibrations satisfying specific conditions, leaving room for further investigation into more general cases. Exploring the properties of the constructed compactifications, such as their singularities and birational geometry, presents avenues for future research. Additionally, investigating the implications of these findings for related areas like mirror symmetry and derived categories could yield fruitful results.

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Compactifying Lagrangian fibrations

by Giul... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06505.pdf

Deeper Inquiries

How do the methods presented in the paper extend to Lagrangian fibrations with more general singular fibers?

The methods presented in the paper heavily rely on the existence of local sections over an open subset of the base whose complement has codimension at least two. This assumption is crucial for constructing the relative Albanese fibration, a key ingredient in the compactification process. Here's a breakdown of the challenges and potential approaches for more general singular fibers:
Challenges:

Existence of Relative Albanese: The construction of the relative Albanese fibration, as presented, depends on the existence of local sections. General singular fibers might not admit such sections, making this construction difficult to apply directly.
Extendability of Holomorphic Forms: The criterion for the existence of a symplectic compactification (Theorem 2.3) requires the holomorphic symplectic form to extend to a smooth compactification. More complex singular fibers could lead to situations where this extendability condition is not met.
Smoothness of Compactification: Even if a compactification exists, the presence of more general singular fibers could obstruct the smoothness of the resulting symplectic variety. The techniques used in the paper to prove smoothness (like those in Theorem 4.1 and 6.13) might not generalize easily.
Potential Approaches:

Partial Compactifications: One approach could be to focus on partial compactifications over open subsets of the base where the fibers are "well-behaved" (e.g., admit local sections). This might allow applying the existing techniques to construct compactifications over these subsets and then studying the global behavior.
Modifications of the Base: Similar to Example 2.6, it might be necessary to consider modifications (like blow-ups) of the base variety B to resolve singularities related to the singular fibers. This could lead to a situation where a compactification exists after a suitable modification of the base.
Alternative Constructions: Exploring alternative methods for constructing symplectic compactifications that are less reliant on the existence of local sections could be fruitful. This might involve techniques from birational geometry, deformation theory, or the study of moduli spaces.

Could the existence of a symplectic compactification provide insights into the structure of the base variety B of the Lagrangian fibration?

Yes, the existence of a symplectic compactification can indeed provide valuable insights into the structure of the base variety B. Here's how:

Singularities of B: As mentioned in the paper, the existence of a symplectic compactification (with an extendable symplectic form) imposes restrictions on the singularities of B. For instance, Matsushita's theorem implies that B must have log terminal singularities. This connection between the existence of a symplectic structure and the singularities of the base is a recurring theme in the study of Lagrangian fibrations.
Structure of the Discriminant Locus: The discriminant locus of the Lagrangian fibration, which corresponds to the points in B over which the fibers are singular, plays a crucial role. The existence and properties of a symplectic compactification can shed light on the geometry and topology of this discriminant locus. For example, the codimension and the types of singularities appearing in the discriminant locus can be related to the properties of the compactification.
Moduli of Lagrangian Fibrations: The study of moduli spaces of Lagrangian fibrations is an active area of research. Understanding which Lagrangian fibrations admit symplectic compactifications could lead to a better understanding of the structure of these moduli spaces. The existence of a compactification might distinguish special sub-loci within the moduli space, potentially with interesting geometric properties.

What are the implications of this research for understanding the mirror symmetry of Lagrangian fibrations and their associated Fukaya categories?

This research has the potential to significantly impact our understanding of mirror symmetry for Lagrangian fibrations and their associated Fukaya categories:

Mirror Symmetry and Compactifications: Mirror symmetry predicts a deep relationship between symplectic geometry and complex geometry. The existence of a symplectic compactification for a Lagrangian fibration suggests the possibility of finding a "mirror" compactification on the mirror side. This mirror compactification would be a complex manifold (or variety) equipped with a special Kähler structure, and its properties would be closely related to the original Lagrangian fibration.
Fukaya Categories and Boundary Conditions: The Fukaya category of a symplectic manifold is a powerful invariant that encodes information about its Lagrangian submanifolds. The presence of a symplectic compactification introduces new boundary conditions for the Lagrangian submanifolds, which in turn can affect the structure of the Fukaya category. Understanding how the Fukaya category changes under compactification could provide insights into the mirror symmetry correspondence.
Homological Mirror Symmetry: Homological mirror symmetry is a more refined version of mirror symmetry that predicts an equivalence of categories between the Fukaya category of a symplectic manifold and the derived category of coherent sheaves on its mirror. The existence of symplectic compactifications could lead to new examples where homological mirror symmetry can be studied and potentially proven. The techniques developed in the paper for constructing and analyzing compactifications could be valuable tools in this endeavor.