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Connectedness of fibers beyond semitoric systems II: ephemeral singular points

Daniele Sepe, Susan Tolman

TL;DR

The paper broadens the class of completely integrable Hamiltonian systems with connected fibers by allowing ephemeral degenerate singular points in a complexity-one setting. It develops a Morse-theoretic reduction on the reduced spaces $\Phi^{-1}(\beta)/T$, linking critical points to tall non-degenerate singularities and to ephemeral points, and proves that fibers are connected if and only if no tall non-degenerate singular point has a hyperbolic block with connected stabilizer (under properness of the moment map). An intrinsic jet-based criterion is established to characterize ephemeral points independently of local models, and a family of examples demonstrates the extension of connected-fiber results beyond previous restrictive hypotheses. The results highlight that ephemeral singularities yield a tractable extension of the theory, significantly expanding the class of integrable systems amenable to Morse-theoretic fiber-connectivity analysis.

Abstract

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 $p \! : \! -q$ resonances and special Lagrangian fibrations. Finally, we construct a family of examples that shows that our main theorem meaningfully extends previous results.

Connectedness of fibers beyond semitoric systems II: ephemeral singular points

TL;DR

The paper broadens the class of completely integrable Hamiltonian systems with connected fibers by allowing ephemeral degenerate singular points in a complexity-one setting. It develops a Morse-theoretic reduction on the reduced spaces , linking critical points to tall non-degenerate singularities and to ephemeral points, and proves that fibers are connected if and only if no tall non-degenerate singular point has a hyperbolic block with connected stabilizer (under properness of the moment map). An intrinsic jet-based criterion is established to characterize ephemeral points independently of local models, and a family of examples demonstrates the extension of connected-fiber results beyond previous restrictive hypotheses. The results highlight that ephemeral singularities yield a tractable extension of the theory, significantly expanding the class of integrable systems amenable to Morse-theoretic fiber-connectivity analysis.

Abstract

In an earlier paper, we proved the connectedness of the fibers of every -dimensional integrable system satisfying both: the action extends the action of an -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 -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 resonances and special Lagrangian fibrations. Finally, we construct a family of examples that shows that our main theorem meaningfully extends previous results.
Paper Structure (7 sections, 24 theorems, 52 equations)

This paper contains 7 sections, 24 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Morse theory on reduced spaces
  3. Non-degenerate singular points
  4. Ephemeral points
  5. Proofs of the main results
  6. An intrinsic characterization of ephemeral points
  7. A family of examples

Key Result

Theorem 1.3

Let $(M,\omega, \Phi)$ be a complexity one $T$-space with a proper moment map. Assume that some fiber of $\Phi$ contains (at least) three distinct orbits $\mathfrak O_1, \mathfrak O_2, \mathfrak O_3$ such that, for all $p \in \mathfrak O_i$: Then there is no integrable system of the form $\left(M, \omega, {\mathfrak t} \times{\mathbb R}, f:= (\Phi, g) \right)$ such that every tall singular point

Theorems & Definitions (53)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Example 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 43 more