Semitoric systems of non-simple type

Palmer, Joseph; Pelayo, Álvaro; Tang, Xiudi

doi:10.1007/s13398-024-01656-2

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

2024

DOI: 10.1007/s13398-024-01656-2

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Semitoric systems of non-simple type

Joseph Palmer,

Álvaro Pelayo,

Xiudi Tang

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2024

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Semitoric Families

Le Floch,

Palmer

2024

Memoirs of the AMS

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Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global S 1 \mathbb {S}^1 -action; these systems were classified by Pelayo and Vũ Ngọc 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 \mathbb {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 \mathbb {S}^1 -action. Furthermore, we introduce natural notions of blowup and blowdown in this context, investigate how semitoric families behave under these operations, and use this to prove that each Hirzebruch surface admits a semitoric family with certain desirable invariants; these families are 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.

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Semitoric Families

Le Floch,

Palmer

2024

Memoirs of the AMS

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Semitoric systems of non-simple type

Cited by 1 publication

References 54 publications

Semitoric Families

Semitoric Families

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