Suppose we are given a regular $2n$-dimensional Lagrangian fibration $\pi : (M,\omega) \to B$ with connected, compact fibers. Then it is well-known (Arnold-Liouville) that each fibre is a Lagrangian torus and $B$ has (integral) affine structure. Moreover, for each $x \in B$ there is a neighbourhood $x \in U \subset B$ such that $(\pi^{-1}(U),\omega) \cong (U \times T^n, dx \wedge dy)$ where $(x,y)$ are local action-angle coordinates ($x$ are coordinates on $U$ and $y$ (mod $1$) are coordinates on the torus $T^n$). My question is, what sort of 2-form on the total space $M$ looks like $C_{ij}dx^i \wedge dy^j$ in these coordinates? Here the $C_{ij}$ are elements of a matrix $C \in GL(n,\mathbb{Z})$ and I want to know how, for a given choice of $C$, to define a 2-form that always looks like this in action-angle coordinates. There are probably different ways of doing so, but I would like a "nice" one possibly involving the affine structure of $B$.


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