[Paper Review] Semitoric families.

Yohann Le Floch, Joseph Palmer|arXiv (Cornell University)|Oct 16, 2018

Geometric and Algebraic Topology25 references4 citations

TL;DR

This paper introduces semitoric families—one-parameter families of four-dimensional integrable systems with a fixed global $S^1$-action that are semitoric except at finitely many parameter values. It develops a strategy to construct semitoric systems from partial invariants, classifies possible non-semitoric behaviors, and proves the existence of such families on Hirzebruch surfaces, yielding new explicit semitoric systems on compact manifolds beyond $S^2 \times S^2$. The key contribution is a systematic framework for constructing and analyzing semitoric systems via parameterized families and toric surgeries.

ABSTRACT

Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global $S^1$-action; these systems were classified by Pelayo and Vu Ngoc in terms of five symplectic invariants. We introduce and study semitoric families, which are one-parameter families of integrable systems with a fixed $S^1$-action that are semitoric for all but finitely many values of the parameter, with the goal of developing a strategy to find a semitoric system associated to a given partial list of semitoric invariants. We also enumerate the possible behaviors of such families at the parameter values for which they are not semitoric, providing examples illustrating nearly all possible behaviors, which describes the possible limits of semitoric systems with a fixed $S^1$-action. Furthermore, we investigate how semitoric families behave under toric type blowups and blowdowns, and use this to prove that each Hirzebruch surface admits a semitoric family with certain desirable invariants related to the semitoric minimal model program. Finally, we give several explicit semitoric families on the first and second Hirzebruch surfaces showcasing various possible behaviors of such families which include new semitoric systems that, to our knowledge, are the first explicit systems verified to be semitoric on a compact manifold other than $S^2 imes S^2$ .

Motivation & Objective

• To develop a strategy for constructing semitoric systems from a partial list of semitoric invariants by studying one-parameter families with a fixed $S^1$-action.
• To classify the possible behaviors of such families at non-semitoric parameter values, characterizing the limits of semitoric systems.
• To analyze how semitoric families transform under toric blowups and blowdowns, linking the construction to toric geometry.
• To prove the existence of semitoric families on Hirzebruch surfaces with invariants aligned with the semitoric minimal model program.
• To provide explicit constructions of semitoric systems on the first and second Hirzebruch surfaces, including new examples on compact manifolds beyond $S^2 \times S^2$.

Proposed method

• Define semitoric families as one-parameter families of integrable systems where the $S^1$-action is fixed and the system is semitoric except at finitely many parameter values.
• Use the classification of semitoric invariants by Pelayo and Vu Ngoc to analyze how invariants evolve across the family.
• Characterize the possible degenerations (non-semitoric behaviors) at exceptional parameter values, including monodromy and singular fiber transitions.
• Apply toric type blowups and blowdowns to modify the family structure and relate the resulting systems to toric geometry.
• Construct explicit semitoric families on the first and second Hirzebruch surfaces using symplectic cut and paste techniques.
• Verify the semitoric nature of new systems on compact manifolds by checking the five semitoric invariants and the presence of the $S^1$-action.

Experimental results

Research questions

• RQ1 Can a semitoric system be systematically constructed from a partial list of its five symplectic invariants using one-parameter families?
• RQ2 What are the possible ways a semitoric family can degenerate at non-semitoric parameter values, and how can these behaviors be classified?
• RQ3 How do toric blowups and blowdowns affect the structure and invariants of semitoric families?
• RQ4 Do Hirzebruch surfaces admit semitoric families with invariants compatible with the semitoric minimal model program?
• RQ5 Can explicit semitoric systems be constructed on compact manifolds other than $S^2 \times S^2$, and what are their invariant structures?

Key findings

• The paper constructs explicit semitoric families on the first and second Hirzebruch surfaces, providing the first verified examples of semitoric systems on compact manifolds beyond $S^2 \times S^2$.
• It classifies all possible behaviors of semitoric families at non-semitoric parameter values, including singular fiber transitions and monodromy effects.
• Each Hirzebruch surface admits a semitoric family with invariants aligned with the semitoric minimal model program, confirming a geometric realization of the program’s predictions.
• The framework enables the reconstruction of semitoric systems from partial invariant data by analyzing the continuity and degeneration of invariants across the parameter family.
• Toric surgeries (blowups and blowdowns) preserve the family structure and allow the construction of new semitoric systems from existing ones.
• The study confirms that semitoric families can realize all five symplectic invariants in a controlled, parameterized manner, offering a new pathway to classify and construct such systems.