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: $$ x = \frac{t}{1-t}, \quad y = \frac{1/t}{1-1/t} = \frac{1}{t - 1} $$ thus $x + y = -1$ OK, good enough.