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 $\eta \in [-1,1], \phi \in [0, 2\pi]$, and we consider $\epsilon e^{\eta + i \phi}$, framing the $\epsilon$-scale annuli.