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--- blog:2023-03-22 [2023/03/23 08:26] – created pzhou
+++ blog:2023-03-22 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 11: / Line 11: @@
 Aha! How about this:
-$$ x = \frac{t}{1-t}, \quad y = \frac{1/t}{1-1/t} = \frac{1}{t - 1} $$
+$$ x = \frac{t}{1+t}, \quad y =   \frac{1}{1+t} $$
-thus $x + y = -1$ OK, good enough.
+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.
+==== 2 dim ====
+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.
+==== non-unimodular case? quotient? ====
+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?