Connectedness of fibers beyond semitoric systems II: ephemeral singular points
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911686640795648 |
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| author | Sepe, Daniele Tolman, Susan |
| author_facet | Sepe, Daniele Tolman, Susan |
| contents | In an earlier paper, we proved the connectedness of the fibers of every $2n$-dimensional integrable system satisfying both: the action extends the action of an $(n-1)$-dimensional torus which has a proper moment map, and every tall singular point is non-degenerate and no such point has a hyperbolic block and connected $T$-stabilizer. Unfortunately, these criteria are fairly restrictive. Our main goal in this paper is to find a larger class of integrable systems that has connected fibers by weakening the non-degeneracy assumption above. To achieve this, we introduce ``ephemeral" degenerate singular points, examples of which have appeared in the literature in the context of both $p \! : \! -q$ resonances and special Lagrangian fibrations. Finally, we construct a family of examples that shows that our main theorem meaningfully extends previous results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Connectedness of fibers beyond semitoric systems II: ephemeral singular points Sepe, Daniele Tolman, Susan Symplectic Geometry Dynamical Systems Primary: 37J35, 53D20. Secondary 37J39, 53D35 In an earlier paper, we proved the connectedness of the fibers of every $2n$-dimensional integrable system satisfying both: the action extends the action of an $(n-1)$-dimensional torus which has a proper moment map, and every tall singular point is non-degenerate and no such point has a hyperbolic block and connected $T$-stabilizer. Unfortunately, these criteria are fairly restrictive. Our main goal in this paper is to find a larger class of integrable systems that has connected fibers by weakening the non-degeneracy assumption above. To achieve this, we introduce ``ephemeral" degenerate singular points, examples of which have appeared in the literature in the context of both $p \! : \! -q$ resonances and special Lagrangian fibrations. Finally, we construct a family of examples that shows that our main theorem meaningfully extends previous results. |
| title | Connectedness of fibers beyond semitoric systems II: ephemeral singular points |
| topic | Symplectic Geometry Dynamical Systems Primary: 37J35, 53D20. Secondary 37J39, 53D35 |
| url | https://arxiv.org/abs/2510.16976 |