Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A smooth integrable system on a symplectic $2n$-manifold $(M,\omega)$ is a map $F=(F_1,\ldots,F_n)$ whose components Poisson commute and have linearly independent differentials at almost every point, with respect to the symplectic volume measure $\omega^n/n!$. The Poisson bracket is $\{u,v\}=\omega(X_u,X_v)$, where $\iota_{X_u}\omega=-du$. The connected components of common level sets form a singular Lagrangian fibration. On a compact regular fiber, choose a cycle basis $\gamma_i$ and a local primitive $d\alpha=\omega$; the action coordinates are $I_i=(2\pi)^{-1}\oint_{\gamma_i}\alpha$, with changes in $GL(n,\mathbb Z)\ltimes\mathbb R^n$. They define the integral affine structure of the regular base. Nondegeneracy means that the transverse quadratic forms of the integrals generate a Cartan subalgebra of the symplectic linear Lie algebra. Their Williamson normal form is a direct sum of elliptic blocks $(q^2+p^2)/2$, hyperbolic blocks $qp$, and focus--focus pairs $(q_1p_2-q_2p_1,q_1p_1+q_2p_2)$. In dimension four an almost toric fibration has only nondegenerate elliptic and focus--focus singularities. The source considers a two-dimensional base domain with boundary and corners, with isolated focus singularities, carrying the action-induced affine structure. An affine equivalence of such bases is a stratified diffeomorphism preserving this singular affine structure; on regular charts it is an integral affine map.

Statement recorded in the supplied source.

For two such almost toric fibrations $\phi:(M^4,\omega)\to B$ and $\phi':({M'}^4,\omega')\to B'$ with compact fibers, does an affine diffeomorphism $B\to B'$ imply that the fibrations are symplectomorphic? Here equivalence of fibrations means a symplectomorphism sending fibers to fibers. [S2]

Short English statement

Does the singular integral affine base determine an almost toric four-dimensional Lagrangian fibration up to symplectic equivalence?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100202 (AMR-020-0202). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Problem 2.2 and preceding almost-toric paragraph. https://arxiv.org/abs/1804.03737.