There are three approaches,
Daping's canonical skeleton using $\omega = Im(\Omega)$.
Constructing Kahler potential by extending (after modification) standard KP on $(\C^*)^n$.
Semi-Tropicalize the cluster variety, put the main torus on some flat region. Let trickle down from
They each need to solve some technical problems.
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The equation for $X_1$ is $x x' = 1 + q$, $q \in \C^*, x,x' \in \C$. We have several approaches to build a Liouville structure on this space. They are related here but have different abilities to generalize to higher dimension.
Build a Kahler potential $\varphi$.
Use symplectic SYZ fibration, singular affine structure in the base.
Non-toric blow-up for $\C \times \C^*$
Use the holomorphic symplectic structure.
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