Pullback Hamiltonians, maximality, and toric Floer resonances

Definitions and mathematical statement

The source reports Polterovich's suggestion that pullback Hamiltonians from a fibration form a "maximal torus" and provide Floer-homology examples. It explicitly leaves the fibration type uncertain and says later locally toric discussion was not recorded. It also has a type error: for $\Phi:M\to B$, the pullback must use $h:B\to{\mathbb{R}}$, not the printed $H:M\to{\mathbb{R}}$. Functions are Hamiltonians, not elements of $\operatorname{Ham}(M)$ until their flows are taken. The quoted torus terminology is therefore not a literal general theorem.

For a precise regular statement, let $(M,\omega)$ be symplectic and $\Phi:M\to B$ a surjective smooth submersion. Put \[ \mathcal A_\Phi=\Phi^*C^\infty(B). \] The aid proves that $\mathcal A_\Phi$ Poisson-commutes if and only if every fiber is coisotropic: for $V=\ker{\,\mathrm{d}}\Phi$, $V^\omega\subset V$. If the fibers are connected Lagrangians, the full Poisson centralizer is \[ \{F\in C^\infty(M):\{F,h\circ\Phi\}=0\text{ for all }h\in C^\infty(B)\} =\mathcal A_\Phi. \] Thus this is a maximal Poisson-commutative algebra, not automatically a finite-dimensional torus group. Completeness is additionally needed to turn every Hamiltonian into a globally defined flow; compactness supplies it in the following case.

Let $M^{2n}$ be compact connected toric symplectic, with effective $T^n={\mathbb{R}}^n/{\mathbb{Z}}^n$ action and moment map $\Phi:M\to\Delta$. Choose the period-one action-angle normalization. For smooth $h$ on the moment polytope, meaning smoothly extendible as required for $h\circ\Phi$, its interior flow is \[ (p,\theta)\longmapsto(p,\theta+t\nabla h(p)). \] All such time-one flows form an infinite-dimensional abelian group. The exact kernel of the map $h\mapsto\varphi_{h\circ\Phi}^1$ consists of \[ h(p)=c+\langle m,p\rangle,\qquad c\in{\mathbb{R}},\ m\in{\mathbb{Z}}^n. \] The affine-linear subfamily $h(p)=c+\langle a,p\rangle$, $a\in{\mathbb{R}}^n$, alone yields the genuine toric $T^n$, a maximal Hamiltonian torus. Its lattice depends on the explicitly chosen period normalization.

On the free interior locus, an entire fiber is a one-periodic orbit family precisely at \[ \nabla h(p)\in{\mathbb{Z}}^n. \] The zero integer vector gives constant trajectories. If such a resonant $p$ is isolated and $\operatorname{Hess}h(p)$ is nonsingular, its fiber is Morse--Bott: the kernel of the linearized time-one map minus the identity is exactly the tangent space of that torus. Degenerate Hessians can require a larger critical-family analysis. Boundary faces have isotropy and are outside this interior criterion. These statements clarify proposed Floer examples but do not compute Floer homology or supply the missing singular-fibration hypotheses.

Short English statement

The corrected pullback algebra commutes precisely for coisotropic fibers and is maximal as a Poisson algebra for connected Lagrangian fibers. In a compact toric example its general flow group is infinite-dimensional; only affine-linear Hamiltonians give the ordinary torus. Integral-gradient resonances describe interior periodic tori for Floer analysis.

Sources

• Local UnsolvedMath registry, permanent ID $20002081$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[418]. Statement status reconstructed_unverified; clean statement null.
• Michael Hutchings, with help from Yasha Eliashberg and John Etnyre, "Holomorphic curves in contact geometry," AIM workshop problem list, version 15 October 2003, Question $1.23$: https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. Polterovich is the suggestion's named source. The supplied verification says PDF and HTML share the type error; neither was opened in this session.
• Supplied research attempt $1$, result context_only, run status partially_solved, rigor label "proved," novelty confidence "low." Registry status "open," classification OPEN-TRIAGE, suggested difficulty L2, updated 14 August 2026. Original remarks and literature are empty; no full authoritative theorem references or proposer/year registry values are supplied.