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Flop Connections Between Minimal Models of Algebraically Integrable Foliations on Potentially KLT Varieties


Core Concepts
This paper proves that different minimal models of algebraically integrable foliations on potentially klt varieties are connected by a sequence of flops, extending previous results for varieties and specific types of foliations.
Abstract

Bibliographic Information:

Chen, Y., Liu, J., & Wang, Y. (2024). Flop between algebraically integrable foliations on potentially klt varieties. arXiv preprint arXiv:2410.05764v1.

Research Objective:

This paper investigates the relationship between different minimal models of algebraically integrable foliations on potentially klt varieties. The main research question is whether these minimal models are connected by a sequence of flops, similar to the established results for varieties.

Methodology:

The authors employ techniques from algebraic geometry, specifically the minimal model program (MMP) for foliations. They adapt Kawamata's proof for the flop connection between minimal models of varieties to the context of algebraically integrable foliations. The proof involves constructing specific MMPs, analyzing discrepancies of divisors, and utilizing properties of generalized foliated quadruples.

Key Findings:

The paper's main finding is a proof that any two minimal models of an lc algebraically integrable foliated triple on potentially klt varieties are connected by a sequence of flops. This result holds for both Q-factorial and non-Q-factorial varieties.

Main Conclusions:

The authors conclude that the connection between minimal models via flops, previously established for varieties, extends to algebraically integrable foliations on potentially klt varieties. This finding contributes to the understanding of the geometry of foliations and the broader minimal model program.

Significance:

This research advances the minimal model program for foliations by establishing a key connection between different minimal models. It provides a deeper understanding of the birational geometry of foliated varieties and opens avenues for further research in this area.

Limitations and Future Research:

The paper primarily focuses on algebraically integrable foliations. Future research could explore whether similar flop connections exist for non-algebraically integrable foliations. Additionally, extending these results to higher-dimensional varieties and more general settings within the minimal model program presents further research opportunities.

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Deeper Inquiries

Can the results of this paper be extended to other classes of foliations beyond algebraically integrable ones?

Yes, the authors suggest that the results can potentially be extended to non-algebraically integrable foliations on threefolds, contingent upon the assumption that the minimal model program (MMP) holds for generalized foliated quadruples in this setting. Here's a breakdown of why this extension is plausible and the challenges involved: Algebraically integrable case: The current paper leverages the well-established MMP for algebraically integrable foliations. This allows them to run MMPs on foliated triples and generalized foliated quadruples, ultimately leading to the connection of minimal models via flops. Non-algebraically integrable case: For general foliations, the MMP is not yet fully established. However, substantial progress has been made in dimension 3, particularly for rank 2 foliations. Conjecture 4.4: This conjecture posits the existence and termination of MMPs for Q-factorial NQC lc generalized foliated quadruples on klt varieties in dimension 3. It also includes specific cases for different ranks and singularity types. Theorem 4.6: This theorem states that if Conjecture 4.4 holds for a specific case (denoted by a subscript λ), then the flop connection result (analogous to Theorems 1.1-1.3) holds for Q-factorial lc foliated triples on klt threefolds in that case. In essence, the paper lays out a roadmap: proving the MMP for generalized foliated quadruples in dimension 3 would automatically extend the flop connection result to a broader class of foliations. This highlights the interconnected nature of these concepts in the study of foliated varieties.

What are the implications of these findings for the classification of foliated varieties?

The results of this paper have significant implications for the classification of foliated varieties, particularly in the context of the Minimal Model Program: Understanding Birational Geometry: The paper establishes a crucial link between different minimal models of a foliated variety. Just as in the classical MMP for varieties, where minimal models are connected by flops and other birational transformations, this work shows that a similar principle applies in the realm of foliations. Uniqueness and Moduli: While minimal models are not unique in general, the fact that they are connected by a specific type of birational surgery (flops) provides a way to relate their properties. This is a fundamental step towards understanding the moduli spaces of foliated varieties, which parametrize their possible birational models. Invariants: The existence of flops connecting minimal models suggests that certain invariants of foliated varieties should be preserved under these transformations. Identifying and studying such invariants is crucial for distinguishing different birational classes of foliated varieties. Building Blocks: The classification of algebraic varieties aims to identify "building blocks," such as minimal models, and understand how more complicated varieties are constructed from them. This paper contributes to this program by providing a clearer picture of the relationships between these building blocks in the foliated setting. Overall, by establishing the flop connection between minimal models, this paper provides a powerful tool for organizing and classifying foliated varieties. It paves the way for a deeper understanding of their birational geometry and the development of a more comprehensive classification theory.

How does the concept of a "flop" in algebraic geometry relate to other geometric transformations, and what insights do these connections offer?

In algebraic geometry, a flop is a specific type of birational surgery that modifies a variety in a controlled way. It's helpful to understand flops in relation to other geometric transformations: Birational Transformations: These are maps between algebraic varieties that are isomorphisms (bijective and structure-preserving) outside of lower-dimensional subsets. They can be thought of as "almost isomorphisms" that allow us to study varieties by looking at their simpler birational models. Blowing Up/Down: A fundamental type of birational transformation. Blowing up replaces a point on a variety with a copy of projective space, while blowing down does the reverse. This process is often used to resolve singularities (make them milder). Flips and Flops: These are more specialized birational transformations used in the Minimal Model Program (MMP). Flips improve the positivity of the canonical divisor (a measure of complexity) by contracting curves on which the canonical divisor has negative intersection number. Flops are similar to flips but are "volume-preserving." They occur when the canonical divisor has zero intersection with the curves being contracted. Insights from Connections: MMP and Classification: Flops play a crucial role in the MMP, which aims to classify algebraic varieties by finding "minimal models" with simpler properties. The fact that minimal models are connected by flops is a key result in this program. Mirror Symmetry: Flops have a fascinating connection to mirror symmetry, a phenomenon in string theory and algebraic geometry. Mirror symmetry relates pairs of Calabi-Yau manifolds (special types of varieties), and it has been observed that flops on one side of the mirror often correspond to more easily understandable geometric transformations on the other side. Derived Categories: Flops also have deep connections to the study of derived categories, which are powerful tools for understanding the structure of algebraic varieties. It has been shown that flops induce equivalences of derived categories, providing a way to relate the homological algebra of different birational models. In summary, flops are not isolated geometric transformations but are deeply intertwined with other concepts in algebraic geometry. Their study provides insights into the classification of varieties, mirror symmetry, and the structure of derived categories.
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