A simple exercise in skeleton. If I cannot do it in 20 minutes, then I go to sleep.

Consider the simplest case, $\C^* \RM \{1\}$. Can you realize its skeleton as a circle of radius $\epsilon$ around $1$, together with some circle attached to it? Let's parametrize $\C^*$ using $e^{\rho + i \theta}$, then we have a Kahler potential, that is periodic in $\theta$.

OK, great. The whole construction would be invariant under $\rho \to -\rho$. And, if we don't care about strictly being a circle, just homeomorphism, then, it is ok.

We can just embed this pair-of-pants into $(\C^*)^2$ using equation $x + y = 1$. Now, how do I map the solution to the equation back to $\C^*$, such that $x \to 0$ means $t \to 0$ and $y \to 0$ means $t \to \infty$.

Aha! How about this: $$ x = \frac{t}{1+t}, \quad y = \frac{1}{1+t} $$

We can put a Kahler metric on this. By first do a semi-tropicalization, then, put the attracting center at $|x| = |y| \gg 1$. Since things are really symmetric, I will get that skeleton, for this case.

This is the case where $B = \R / \Z = S^1$, and $H = \{0\} \In B$. Then, $B_{\C} = \C^*$. Fine.

No, not fine. I want something not like this. I want to say, my skeleton is $B$. And then I made an infinitesimal, very much local, real blow-up.

I want to change my Liouville form only locally. How should I change my $\varphi$ (which was previously just $\rho^2$.) What if I add to $\varphi$, a compactly supported function $\eta( r)$,

OK, this is actually pretty good. Let $\eta \in [-R,R], \phi \in [0, 2\pi]$, and we consider $\epsilon e^{\eta + i \phi}$, framing the $\epsilon$-scale annuli. We have, where $\eta=1$, a function which is $[\epsilon e^\eta \cos(\phi)]^2$. Then, we can extend that to a function of the form $\eta^2 - C$ near $\eta=0$. OK, do-able. How do you put a puncture?

Say you used to have $\omega = dx dy$, now you need to change it to $(1 + F(r)) dx dy$. You previously have $\lambda = y dx$, now you need to change it to $F(r) r dr d \theta = d ( G(r) d \theta) $ where $G(r) d \theta$ is a well-defined 1-form.

OK, I think this can be done in a local way.

Of course, the simplest way is to do a product. What's the local model? $\R^2$, $T^*\R^2$. What's the local model? Cylinder with two stops at one end. The neighborhood and thickening is nice.

What happens, if you have 3 lines come together? Is there some parameter? I don't think so. Here is the reason. You can define skeleton basically by erecting a conormal, then put a lollipop at the end. There is no ambiguity at putting a lollipop. And if you have several such lollipop, these lollipops are disjoint.

How to describe the upstair space $(t-1)^2 = uv$, and $(t-1) = (uv)^2$, and $t^2 - 1 = uv$? Maybe the first one is more serious. You can deform it, to $(t-1)(t-1-\epsilon)=uv$, or you can resolve it. I guess, we want to deform it.

suppose we have several copies of this space, and we want to do multiplicative reduction, what do we do? We consider certain product, like $t_1 t_2 t_3^2 = b$ (say we want to do weighted proj space), and then we quotient out the fiber by something.

Downstairs space, what do we do with the quotient step?