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Article Dans Une Revue Memoirs of the American Mathematical Society Année : 2021

Semitoric families

Résumé

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 $S^2 \times S^2$ .
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Dates et versions

hal-01895250 , version 1 (14-10-2018)
hal-01895250 , version 2 (14-01-2019)

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Yohann Le Floch, Joseph Palmer. Semitoric families. Memoirs of the American Mathematical Society, In press. ⟨hal-01895250v2⟩
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