====== 2025-01-15 ======

I have been thinking about Liouville sector. The condition on stop is very harsh, in the sense that the Liouville flow need to preserve the boundary.  I am not sure if GPS themselves constructed these required structures. 
  * Construct Liouville structure? 
  * Examples

===== Liouville structure =====

On the other hand, it is possible to develop a Liouville structure on $\C_z$, where the volume form is like $\frac{r^2}{1+r^2} r dr d\theta$.It vanishes near $r=0$, but is otherwise perfectly fine. The reason we want a vanishing volume form is because, when we pushforward by $w=z^2$, let $\rho = r^2$, then it become normal, $\rho / (1+ \rho} d\rho d \psi$, $\psi = 2\theta$. The area form looks like a 'cigar'. Or some other way to interpolate between the standard symplectic form on $\C_z$, and radial flow near the center. 

===== Examples =====

Let's think about $Sym^2(\C)$, what happens if $x_1=x_2$, but $y_1 \neq y_2$? Do we pretend that nothing bad happens? Or we do something special? Well, I want to cut it out. I want to put a Liouville sector structure on the open subset $|x_1 - x_2| < \delta$. The actual diagonal is deep inside. I will use the product Liouville form away from the complex diagonal, do the Perutz patching near the diagonal. So it should work. 

Now, inside this sector near the diagonal, we have a free translational symmetry, so I will use center of mass coordinate $x_1 + x_2$, to do a further cut. The Perutz patching is in a different direction. The whole thing is a bundle over the center of mass coordinate. 

OK, so out of the $1/3$ radius in the tube, I want to slowly transition the function from $(x_1+x_2)/2$, to $max(x_1, x_2)$. It can be done. The transition is outside the Perutz patching region, so we are in the product structure region, everything is fine and nice, so we are done. 

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Now, let's consider $Sym^3(\C)$. Now the diagonal have two strata, one is everything are together, and one is only two are together. I am not sure how Perutz patching is doing, but I guess, deep in the $2+1$ loci, it should be a product structure, yea.