====== Skeleton for framed $A_1$ quiver ======
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. 

===== Auroux's potential (no) =====
$\gdef\ddbar{\d \bar{\d}}$
Consider the fibration
$$ \pi: X_1 \to \R^2, \quad \pi = (b_1,b_2)=(\log|q|, |x|^2 - |x'|^2). $$
It has singular fiber over $b_1=0, b_2=0$, i.e. $|q|=1, |x|=|x'|$. 

A Kahler potential $\varphi$ is a pluri-subharmonic function, meaning $i \ddbar \varphi$ is a $2$-form that integrate over any germ of holomorphic disk is positive. Basic psh function on $\C$ is a function with $\Delta(\varphi)>0$, for example if $z=x+iy$, then $x^2+y^2$ is convex hence fine, or $x^2$ or even $2x^2-y^2$ is fine as well. Also if $f: X \to Y$ is holomorphic, and $\varphi: Y \to \R$ is psh, then $f^* \varphi$ is psh. 

** Oops, I think Gammage-Le made a mistake in claiming some function is psh **

If we let 
$$ \Phi_c = (\log |xy-1| - c)^2 + (|x|^2 - |y|^2)^2 $$
then its Levi matrix $\partial_{z_i} \bar\partial_{z_j} \Phi_c$ is
<nowiki>
$$
L_{\Phi_c}
=
\frac{1}{2|xy-1|^2}
\begin{pmatrix}
|y|^2 & y\bar{x} \\
x\bar{y} & |x|^2
\end{pmatrix}
+
\begin{pmatrix}
4|x|^2 - 2|y|^2 & -2\bar{x}y \\
-2x\bar{y} & -2|x|^2 + 4|y|^2
\end{pmatrix}.
$$
</nowiki>
which is not a hermitian semi-positive definite matrix everywhere. 

For example, when $y=-1, x=0$, we get
<nowiki>
$$ 
\begin{pmatrix}
-3/2 & 0 \\
0 & +4
\end{pmatrix}.
$$
</nowiki>

===== New Construction, using Tailoring =====

We revisit the space $x y = 1 + q$. We want to modify the equation so that, when $|q| \ll 1$, we approximate $1+q \approx 1$, and when $|q| gg 1$, we approximate $1+q \approx q$. That is a new operation called $\tilde{+}$. 

So we deform the defining equation to $xy = 1 \tilde{+} q$. We want to put the skeleton at $|q| = r \ll 1$. So we put down a term like
$$  (\log|q| - \log(r))^2. $$
Then, when $|q|$ is there, we also want to $x, y$ to concentrate at modulus $1$. Since at the small $q$ region, we already decoupled $x,y$ with $q$, we only have $xy=1$, we can use the Kahler potential $|x|^2 + |y|^2$, nice and balanced. 


So, to get the nice skeleton, we use Kahler potential
$$ \varphi = (\log (|q| / r) )^2 + |x|^2 + |y|^2, \quad 0 \ll r \ll 1 $$
but we deform the embedded 'hypersurface' to 
$$ xy = 1 \tilde{+} q. $$
We define $\chi(t)$ a smooth cutoff function that vanish for $t<1/2$ and $=1$ when $t>=1$. We write $\chi_\epsilon(t) = \chi(t/\epsilon)$. Then we define
$$ a \tilde + b = (1-\rho)(a+b) + \rho \chi_\epsilon( |a|^2 / (|a|^2 + |b|^2)) a + \chi_\epsilon( |b|^2 / (|a|^2 + |b|^2)) b. $$
where $\rho_\delta(r)$ is another cut-off function, depending on $r=|a|^2+|b|^2$, and is $1$ when $r > \delta$. We don't need to use it here, since $1$ has a fixed nonzero scale, but we may need it later. 

===== Symplectic SYZ fibration =====
$\gdef\La{\Lambda}$
We first recall the  integral affine structure on a manifold $B$. And integral affine structure
is a local system of lattice $\La_B \In TB$ over $B$, such that $\La_B \otimes_\Z \R = TB$. Of course, $\La^*_B \In T^*B$ satisfies $\La^*_B \otimes_\Z \R = T^*B$. 

Given an integral affine structure on $B$, we may form a canonical Lagrangian torus bundle by 
$$ S_B:= T^*B / \La^*_B. $$
We may locally choose standard coord chart on $B$, $B \supset U \into \R^n$, where $\R^n$ has the standard integral affine structure. 

Similarly, we can build a canonical complex manifold. If $B$ has a connection on $TB$, then we can build a almost complex structure $J$ on $TB$, since the connection gives a splitting 
$$ T_{(b,f)}(TB) = H_{(b,f)}(TB) \oplus V_{(b,f)}(TB) \cong T_b B \oplus T_b B $$
The local system given by the lattice gives such a connection. Such complex structure is integrable. 

Here is a description of $B^o$, where 
$$ B^o = \{ x> 0\} \cup \{x \leq 0, y > 0\} \cup \{x \leq 0, y < x \} / \sim, \quad  (-a, 0) \sim (-a,-a), \forall a > 0$$
Now on $B^o$ we have the induced integral structure of $\R^2$, namely  at every place, we have $\Z \d_x \oplus \Z \d_y$ in the tangent fiber. 

Then, upon the identification along the cut, I want to identify the tangent space along the upper bank $+$, and lower bank $-$ 
$$ \d_x^+ = (\d_x + \d_y)^-, \quad \d_y^+ = \d_y^-. $$
The first equation has no choice, and basically is determined by gluing along the cut; the 2nd equation is where interesting things happens. 

Now, how about the cotangent bundle identifications? We use global prequotient basis $\Z dx \oplus \Z dy$,
then we identify
$$ dy^+ = (dy - dx)^-, \quad dx^+ = dx^- $$

yeah, instead of writing the pretty unipotent monodromy matrix, here we are more explicit. 

We consider how the Lagrangian torus coordinates are identified. Previously, in $T^*\R^2$, we use $\sum_i \theta_i dx_i$, where $\theta_i$ are fiber coordinates. Now, we have
$$ (\theta_x dx + \theta_y dy)^+ = (\theta_x dx + \theta_y dy)^- $$
that unpackages to 
$$ \theta_x^- =\theta_x^+ - \theta_y^+, \quad \theta_y^- = \theta_y^+. $$
Indeed, linear function of the cotangent fiber behave like tangent vector. 

Geometrically, a (unoriented) hyperplane in the base has a 1-dim conormal direction, that become a fiberwise 1-dim circle. That is not the isotropic circle which shrinks to a point at the singular fiber. 
Instead, we are looking for a global primitive section of the integral affine cotangent lattice. In this case, we have it, it is $dx^+ = dx^-$. 

Now, on the glued up space, we need to decide how to fill in the central fiber. Or more precisely, given Auroux's original symplectic form, induced from the standard Kahler form on $\C^2 \times \C^*$, how to deduce Auroux's integral affine structure on the base? 


==== Integral Affine Structure from Auroux ====

Suppose we fix $|q| = r$ and $|x|^2 - |y|^2 = b > 0$. The torus can be described by $\theta_x$ and $\theta_q$. Somehow, we can use the Lagrangian torus fibration of 
$$ \C^2 \times \C^* \to (\R_+)^2 \times \R, \quad (x, y, q) \mapsto (|x|^2, |y|^2, \log q) $$

Pick a point $(r_1, r_2, l) \in (\R_+)^2 \times \R$, and pick a linear 1-form on the fiber, we get a tangent vector in the base. For example, if I say $d(\theta_x + \theta_y)$ that 1-form translates into $\d_{r_1} + \d_{r_2}$. 


Now, we pass to the subset $X_1 \In \C^2 \times \C^*$, its shadow in $\R_+^2 \times \R$ is something, like an amoeba. Indeed, for a given $|q| = e^l$ value, the range of $|x|^2 |y|^2$ is bounded by triangle inequality
$$ (1 - e^l)^2 \leq |x|^2 |y|^2 \leq (1 + e^l)^2 $$
If we rotate $\theta_q$, that will make $|x|$ and $|y|$ osciallate. To make such oscillation invisible, we map down to $|x|^2 - |y|^2$. 

Now, we can ask, how does the integral affine structure work. Given a torus fiber 
$$  \log|q| = l, \quad |x|^2 - |y|^2 = b $$
We want to know integral 1-form in the torus fiber can be translated into what motion on the base. A 1-form on a 2-torus is a circle foliation. One circle foliation is given by, keeping $\theta_q$ fixed. In the ambient space, $d\theta_q$ translates to $\d_l$. It is a Lag with 2-tori fiber and 1-dim base. That's not quite what the question we want to ask. On that subspace, $d \theta_q$