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Classifying affine structures with focus-focus singularities

Xiudi Tang

Abstract

We study the singular affine structures of integrable systems with focus-focus singular fibers on the image of momentum maps. The classification of singular affine structures is equivalent to the classification of simple semitoric systems up to fiber-preserving symplectomorphisms but is not equivalent for semitoric systems with multiple pinched fibers and we give counterexamples for any number of more than one pinch on each fiber.

Classifying affine structures with focus-focus singularities

Abstract

We study the singular affine structures of integrable systems with focus-focus singular fibers on the image of momentum maps. The classification of singular affine structures is equivalent to the classification of simple semitoric systems up to fiber-preserving symplectomorphisms but is not equivalent for semitoric systems with multiple pinched fibers and we give counterexamples for any number of more than one pinch on each fiber.
Paper Structure (12 sections, 21 theorems, 49 equations)

This paper contains 12 sections, 21 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Integrable systems with focus-focus fibers
  3. Tropical affine manifolds with singularities
  4. Tropical affine structures
  5. Local symplectic and affine normal forms
  6. Smooth isomorphisms of the singular affine structure
  7. Multi-valued functions and infinite jets
  8. Main results
  9. Fibers with one pinched point
  10. First-order invariants
  11. Fibers with multiple pinched points
  12. Counterexamples

Key Result

theorem 6

An orbit of semi-local germs at a focus-focus fiber under the action of $\mathfrak{V}_+$ is one-to-one correspondent to a focus-focus label of the same multiplicity.

Theorems & Definitions (63)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5: PeTa2018
  • theorem 6: PeTa2018
  • theorem 7: PeTa2018
  • definition 8
  • definition 9
  • definition 10: 2019arXiv190903501v3P
  • ...and 53 more